Laplace Transform
This comprehensive collection of 640+ Laplace Transform MCQs with detailed answers is designed to help students, educators, and professionals master every aspect of the topic. Covering basics, properties, inverse transforms, convolution, unit step and impulse functions, differential equations, control systems, Fourier connections, engineering applications, and advanced mixed problems, it serves as a complete guide for exam preparation and practical learning. Whether you are preparing for competitive exams, engineering courses, or seeking a deeper understanding of applied mathematics, this resource ensures clarity, coverage, and confidence in tackling Laplace Transform challenges.
A) 1
B) s
C) 1/s ✅
D) e^s
Explanation: L{1} = ∫₀^∞ e^(-st) dt = 1/s
Q2. The Laplace transform of f(t) = e^(at) is:
A) 1/(s-a) ✅
B) 1/(s+a)
C) s/(s^2+a^2)
D) a/(s^2+a^2)
Explanation: L{e^(at)} = ∫₀^∞ e^(-(s-a)t) dt = 1/(s-a)
Q3. The Laplace transform of f(t) = sin(at) is:
A) s/(s^2+a^2)
B) a/(s^2+a^2) ✅
C) 1/(s-a)
D) 1/(s+a)
Explanation: L{sin(at)} = a/(s^2+a^2)
Q4. The Laplace transform of f(t) = cos(at) is:
A) s/(s^2+a^2) ✅
B) a/(s^2+a^2)
C) 1/(s-a)
D) 1/(s+a)
Explanation: L{cos(at)} = s/(s^2+a^2)
Q5. Which condition ensures the existence of Laplace transform?
A) f(t) must be continuous
B) f(t) must be of exponential order ✅
C) f(t) must be periodic
D) f(t) must be bounded
Explanation: Laplace transform exists if f(t) is piecewise continuous and of exponential order.
Q6. The Laplace transform of f(t) = t is:
A) 1/s
B) 1/s^2 ✅
C) s/(s^2+a^2)
D) a/(s^2+a^2)
Explanation: L{t} = ∫₀^∞ t e^(-st) dt = 1/s^2
Q7. The Laplace transform of f(t) = t^n is:
A) n!/s^(n+1) ✅
B) 1/s^n
C) s^n/n!
D) (n+1)!/s^n
Explanation: L{t^n} = n!/s^(n+1)
Q8. The Laplace transform of δ(t) (Dirac delta function) is:
A) 1 ✅
B) 0
C) s
D) 1/s
Explanation: L{δ(t)} = ∫₀^∞ e^(-st) δ(t) dt = 1
Q9. The Laplace transform of u(t) (unit step function) is:
A) 1/s ✅
B) s
C) 0
D) 1
Explanation: u(t)=1 for t≥0, so L{u(t)}=1/s
Q10. The Laplace transform of f(t) = e^(-at) is:
A) 1/(s+a) ✅
B) 1/(s-a)
C) s/(s^2+a^2)
D) a/(s^2+a^2)
Explanation: L{e^(-at)} = 1/(s+a)
Q11. The Laplace transform of f(t) = cosh(at) is:
A) s/(s^2-a^2) ✅
B) a/(s^2+a^2)
C) 1/(s-a)
D) 1/(s+a)
Explanation: L{cosh(at)} = s/(s^2-a^2)
Q12. The Laplace transform of f(t) = sinh(at) is:
A) a/(s^2-a^2) ✅
B) s/(s^2+a^2)
C) 1/(s-a)
D) 1/(s+a)
Explanation: L{sinh(at)} = a/(s^2-a^2)
Q13. The Laplace transform of f(t) = e^(at) sin(bt) is:
A) b/[(s-a)^2+b^2] ✅
B) s/[(s-a)^2+b^2]
C) 1/(s-a)
D) 1/(s+b)
Explanation: L{e^(at) sin(bt)} = b/[(s-a)^2+b^2]
Q14. The Laplace transform of f(t) = e^(at) cos(bt) is:
A) (s-a)/[(s-a)^2+b^2] ✅
B) b/[(s-a)^2+b^2]
C) 1/(s-a)
D) 1/(s+b)
Explanation: L{e^(at) cos(bt)} = (s-a)/[(s-a)^2+b^2]
Q15. The Laplace transform of f(t) = t·e^(at) is:
A) 1/(s-a)^2 ✅
B) 1/(s+a)^2
C) s/(s^2+a^2)
D) a/(s^2+a^2)
Explanation: L{t e^(at)} = 1/(s-a)^2
Q16. The Laplace transform of f(t) = t·sin(at) is:
A) 2as/(s^2+a^2)^2 ✅
B) a/(s^2+a^2)
C) s/(s^2+a^2)
D) 1/s^2
Explanation: L{t sin(at)} = 2as/(s^2+a^2)^2
Q17. The Laplace transform of f(t) = t·cos(at) is:
A) (s^2-a^2)/(s^2+a^2)^2 ✅
B) s/(s^2+a^2)
C) a/(s^2+a^2)
D) 1/s^2
Explanation: L{t cos(at)} = (s^2-a^2)/(s^2+a^2)^2
Q18. The Laplace transform of f(t) = e^(at) t^n is:
A) n!/(s-a)^(n+1) ✅
B) 1/(s-a)
C) s^n/n!
D) (n+1)!/s^n
Explanation: L{t^n e^(at)} = n!/(s-a)^(n+1)
Q19. The Laplace transform of f(t) = sin(at)/t is:
A) tan⁻¹(a/s) ✅
B) a/(s^2+a^2)
C) s/(s^2+a^2)
D) 1/s
Explanation: L{sin(at)/t} = tan⁻¹(a/s)
Q20. The Laplace transform of f(t) = cos(at)/t is:
A) ln(s/a) ✅
B) s/(s^2+a^2)
C) a/(s^2+a^2)
D) 1/s
Explanation: L{cos(at)/t} = ln(s/a)
Q21. The Laplace transform is a ______ operator.
A) Linear ✅
B) Non-linear
C) Exponential
D) Random
Explanation: L{af(t)+bg(t)} = aL{f(t)} + bL{g(t)} → linearity.
Q22. If L{f(t)} = F(s), then L{e^(at) f(t)} = ?
A) F(s-a) ✅
B) F(s+a)
C) sF(s)
D) F(s)/s
Explanation: Multiplication by e^(at) shifts the transform: F(s-a).
Q23. If L{f(t)} = F(s), then L{f(t-a) u(t-a)} = ?
A) e^(-as) F(s) ✅
B) F(s-a)
C) F(s+a)
D) sF(s)
Explanation: Time shifting property: L{f(t-a)u(t-a)} = e^(-as)F(s).
Q24. If L{f(t)} = F(s), then L{f(at)} = ?
A) (1/a) F(s/a) ✅
B) aF(s)
C) F(as)
D) F(s-a)
Explanation: Scaling property: L{f(at)} = (1/a)F(s/a).
Q25. L{df/dt} = ?
A) sF(s) - f(0) ✅
B) sF(s)
C) F(s)/s
D) F(s) + f(0)
Explanation: Differentiation property: L{f'(t)} = sF(s) - f(0).
Q26. L{d²f/dt²} = ?
A) s²F(s) - sf(0) - f'(0) ✅
B) s²F(s)
C) F(s)/s²
D) F(s) + f(0)
Explanation: Second derivative property.
Q27. L{∫₀^t f(Ï„) dÏ„} = ?
A) F(s)/s ✅
B) sF(s)
C) F(s-a)
D) e^(-as)F(s)
Explanation: Integration property: Laplace of integral = F(s)/s.
Q28. L{t f(t)} = ?
A) -dF(s)/ds ✅
B) F(s)/s
C) sF(s)
D) F(s-a)
Explanation: Multiplication by t corresponds to differentiation in s-domain.
Q29. L{t^n f(t)} = ?
A) (-1)^n d^nF(s)/ds^n ✅
B) n!F(s)
C) F(s)/s^n
D) s^nF(s)
Explanation: Generalized property: L{t^n f(t)} = (-1)^n d^nF(s)/ds^n.
Q30. L{f(t)/t} = ?
A) ∫_s^∞ F(σ) dσ ✅
B) F(s)/s
C) sF(s)
D) F(s-a)
Explanation: Division by t corresponds to integration in s-domain.
Q31. Convolution property: L{f(t)*g(t)} = ?
A) F(s)G(s) ✅
B) F(s)+G(s)
C) F(s)/G(s)
D) F(s-a)
Explanation: Convolution in time → multiplication in s-domain.
Q32. L{f(t) cos(at)} = ?
A) (F(s-a)+F(s+a))/2 ✅
B) F(s-a)
C) F(s+a)
D) F(s)/s
Explanation: Modulation property with cosine.
Q33. L{f(t) sin(at)} = ?
A) (F(s-a)-F(s+a))/(2i) ✅
B) F(s-a)
C) F(s+a)
D) F(s)/s
Explanation: Modulation property with sine.
Q34. Initial value theorem: f(0+) = ?
A) lim_{s→∞} sF(s) ✅
B) lim_{s→0} F(s)
C) lim_{s→∞} F(s)
D) lim_{s→0} sF(s)
Explanation: Initial value theorem states f(0+) = lim_{s→∞} sF(s).
Q35. Final value theorem: f(∞) = ?
A) lim_{s→0} sF(s) ✅
B) lim_{s→∞} F(s)
C) lim_{s→0} F(s)
D) lim_{s→∞} sF(s)
Explanation: Final value theorem states f(∞) = lim_{s→0} sF(s).
Q36. L{f'(t)} property is useful for:
A) Solving differential equations ✅
B) Solving algebraic equations
C) Finding integrals
D) Finding limits
Explanation: Converts differential equations into algebraic equations.
Q37. L{δ(t-a)} = ?
A) e^(-as) ✅
B) 1
C) 0
D) 1/s
Explanation: Laplace of shifted delta = e^(-as).
Q38. L{u(t-a)} = ?
A) e^(-as)/s ✅
B) 1/s
C) s
D) 0
Explanation: Unit step shifted: L{u(t-a)} = e^(-as)/s.
Q39. If L{f(t)} = F(s), then L{f(t)/a} = ?
A) (1/a)F(s) ✅
B) F(s-a)
C) F(s+a)
D) sF(s)
Explanation: Division by constant scales the transform.
Q40. If L{f(t)} = F(s), then L{af(t)} = ?
A) aF(s) ✅
B) F(s)/a
C) F(s-a)
D) F(s+a)
Explanation: Multiplication by constant scales the transform.
Q41. Find the inverse Laplace of F(s) = 1/s.
A) 1 ✅
B) e^t
C) t
D) δ(t)
Explanation: L⁻¹{1/s} = 1 (unit step function).
Q42. Find the inverse Laplace of F(s) = 1/(s-a).
A) e^(at) ✅
B) e^(-at)
C) cos(at)
D) sin(at)
Explanation: L⁻¹{1/(s-a)} = e^(at).
Q43. Find the inverse Laplace of F(s) = 1/(s+a).
A) e^(-at) ✅
B) e^(at)
C) cos(at)
D) sin(at)
Explanation: L⁻¹{1/(s+a)} = e^(-at).
Q44. Find the inverse Laplace of F(s) = a/(s^2+a^2).
A) sin(at) ✅
B) cos(at)
C) e^(at)
D) t
Explanation: L⁻¹{a/(s^2+a^2)} = sin(at).
Q45. Find the inverse Laplace of F(s) = s/(s^2+a^2).
A) cos(at) ✅
B) sin(at)
C) e^(at)
D) t
Explanation: L⁻¹{s/(s^2+a^2)} = cos(at).
Q46. Find the inverse Laplace of F(s) = 1/s^2.
A) t ✅
B) 1
C) e^(at)
D) sin(at)
Explanation: L⁻¹{1/s^2} = t.
Q47. Find the inverse Laplace of F(s) = 2/(s^3).
A) t^2 ✅
B) t
C) 1
D) e^(at)
Explanation: L⁻¹{2/s^3} = t^2.
Q48. Find the inverse Laplace of F(s) = n!/s^(n+1).
A) t^n ✅
B) t^(n+1)
C) e^(at)
D) sin(at)
Explanation: L⁻¹{n!/s^(n+1)} = t^n.
Q49. Find the inverse Laplace of F(s) = 1/(s^2-1).
A) sinh(t) ✅
B) cosh(t)
C) e^t
D) sin(t)
Explanation: L⁻¹{1/(s^2-1)} = sinh(t).
Q50. Find the inverse Laplace of F(s) = s/(s^2-1).
A) cosh(t) ✅
B) sinh(t)
C) e^t
D) sin(t)
Explanation: L⁻¹{s/(s^2-1)} = cosh(t).
Q51. Find the inverse Laplace of F(s) = 1/(s^2+4).
A) (1/2)sin(2t) ✅
B) cos(2t)
C) e^(2t)
D) t
Explanation: L⁻¹{1/(s^2+4)} = (1/2)sin(2t).
Q52. Find the inverse Laplace of F(s) = s/(s^2+4).
A) cos(2t) ✅
B) sin(2t)
C) e^(2t)
D) t
Explanation: L⁻¹{s/(s^2+4)} = cos(2t).
Q53. Find the inverse Laplace of F(s) = 1/(s^2+9).
A) (1/3)sin(3t) ✅
B) cos(3t)
C) e^(3t)
D) t
Explanation: L⁻¹{1/(s^2+9)} = (1/3)sin(3t).
Q54. Find the inverse Laplace of F(s) = s/(s^2+9).
A) cos(3t) ✅
B) sin(3t)
C) e^(3t)
D) t
Explanation: L⁻¹{s/(s^2+9)} = cos(3t).
Q55. Find the inverse Laplace of F(s) = 1/(s^2+1).
A) sin(t) ✅
B) cos(t)
C) e^t
D) t
Explanation: L⁻¹{1/(s^2+1)} = sin(t).
Q56. Find the inverse Laplace of F(s) = s/(s^2+1).
A) cos(t) ✅
B) sin(t)
C) e^t
D) t
Explanation: L⁻¹{s/(s^2+1)} = cos(t).
Q57. Find the inverse Laplace of F(s) = 1/(s^2+ω^2).
A) (1/ω)sin(ωt) ✅
B) cos(ωt)
C) e^(ωt)
D) t
Explanation: L⁻¹{1/(s^2+ω^2)} = (1/ω)sin(ωt).
Q58. Find the inverse Laplace of F(s) = s/(s^2+ω^2).
A) cos(ωt) ✅
B) sin(ωt)
C) e^(ωt)
D) t
Explanation: L⁻¹{s/(s^2+ω^2)} = cos(ωt).
Q59. Find the inverse Laplace of F(s) = 1/(s^2-ω^2).
A) (1/ω)sinh(ωt) ✅
B) cosh(ωt)
C) e^(ωt)
D) t
Explanation: L⁻¹{1/(s^2-ω^2)} = (1/ω)sinh(ωt).
Q60. Find the inverse Laplace of F(s) = s/(s^2-ω^2).
A) cosh(ωt) ✅
B) sinh(ωt)
C) e^(ωt)
D) t
Explanation: L⁻¹{s/(s^2-ω^2)} = cosh(ωt).
Q61. Convolution theorem states:
A) L{f(t)*g(t)} = F(s)G(s) ✅
B) L{f(t)*g(t)} = F(s)+G(s)
C) L{f(t)*g(t)} = F(s)/G(s)
D) L{f(t)*g(t)} = F(s-a)
Explanation: Convolution in time corresponds to multiplication in s-domain.
Q62. The convolution of f(t) and g(t) is defined as:
A) ∫₀^t f(Ï„)g(t-Ï„)dÏ„ ✅
B) ∫₀^∞ f(t)g(t)dt
C) f(t)g(t)
D) ∫₀^t f(t)g(Ï„)dÏ„
Explanation: Convolution integral definition.
Q63. L⁻¹{F(s)G(s)} = ?
A) f(t)*g(t) ✅
B) f(t)+g(t)
C) f(t)/g(t)
D) f(t)g(t)
Explanation: Inverse Laplace of product is convolution.
Q64. Convolution theorem is useful for:
A) Finding inverse Laplace transforms ✅
B) Solving integrals
C) Solving algebraic equations
D) Finding limits
Explanation: Helps compute inverse Laplace transforms.
Q65. L{sin(t)*cos(t)} = ?
A) (s)/(s^4+5s^2+4) ✅
B) 1/s
C) cos(t)
D) sin(t)
Explanation: Apply convolution theorem.
Q66. Laplace transform is widely used in:
A) Solving differential equations ✅
B) Probability
C) Geometry
D) Statistics
Explanation: Converts differential equations into algebraic equations.
Q67. Laplace transform is used in electrical engineering to:
A) Analyze circuits ✅
B) Measure resistance
C) Find current directly
D) Measure voltage
Explanation: Used for circuit analysis.
Q68. Laplace transform is used in control systems to:
A) Study stability ✅
B) Measure temperature
C) Find pressure
D) Measure mass
Explanation: Helps analyze system stability.
Q69. Laplace transform is used in mechanical engineering to:
A) Study vibrations ✅
B) Measure length
C) Find area
D) Measure density
Explanation: Used for vibration analysis.
Q70. Laplace transform is used in probability to:
A) Find moment generating functions ✅
B) Find mean
C) Find variance
D) Find mode
Explanation: Laplace transform relates to MGF.
Q71. Laplace transform of unit impulse δ(t) is:
A) 1 ✅
B) 0
C) s
D) 1/s
Explanation: L{δ(t)} = 1.
Q72. Laplace transform of shifted impulse δ(t-a) is:
A) e^(-as) ✅
B) 1
C) 0
D) 1/s
Explanation: L{δ(t-a)} = e^(-as).
Q73. Laplace transform of unit step u(t) is:
A) 1/s ✅
B) s
C) 0
D) 1
Explanation: L{u(t)} = 1/s.
Q74. Laplace transform of shifted step u(t-a) is:
A) e^(-as)/s ✅
B) 1/s
C) s
D) 0
Explanation: L{u(t-a)} = e^(-as)/s.
Q75. Laplace transform of periodic function f(t) with period T is:
A) (1/(1-e^(-sT))) ∫₀^T e^(-st)f(t)dt ✅
B) ∫₀^∞ e^(-st)f(t)dt
C) F(s-a)
D) F(s+a)
Explanation: Formula for periodic functions.
Q76. Laplace transform of rectangular pulse of width T is:
A) (1-e^(-sT))/s ✅
B) 1/s
C) e^(-sT)
D) s
Explanation: Rectangular pulse transform.
Q77. Laplace transform of ramp function f(t)=t·u(t) is:
A) 1/s^2 ✅
B) 1/s
C) s
D) e^s
Explanation: Ramp function transform.
Q78. Laplace transform of exponential decay f(t)=e^(-at)u(t) is:
A) 1/(s+a) ✅
B) 1/(s-a)
C) s/(s^2+a^2)
D) a/(s^2+a^2)
Explanation: Exponential decay transform.
Q79. Laplace transform of impulse response h(t) gives:
A) Transfer function ✅
B) Step response
C) Ramp response
D) Frequency response
Explanation: Laplace of impulse response = transfer function.
Q80. Laplace transform is especially useful for:
A) Linear time-invariant systems ✅
B) Non-linear systems
C) Random systems
D) Quantum systems
Explanation: Laplace transform applies to LTI systems.
Q81. L{u(t)} = ?
A) 1/s ✅
B) s
C) 0
D) 1
Explanation: Unit step transform is 1/s.
Q82. L{u(t-a)} = ?
A) e^(-as)/s ✅
B) 1/s
C) s
D) 0
Explanation: Shifted unit step transform.
Q83. L{δ(t)} = ?
A) 1 ✅
B) 0
C) s
D) 1/s
Explanation: Impulse transform is 1.
Q84. L{δ(t-a)} = ?
A) e^(-as) ✅
B) 1
C) 0
D) 1/s
Explanation: Shifted impulse transform.
Q85. L{rectangular pulse of width T} = ?
A) (1-e^(-sT))/s ✅
B) 1/s
C) e^(-sT)
D) s
Explanation: Rectangular pulse transform.
Q86. L{triangular pulse of width T} = ?
A) (1-e^(-sT))^2/s^2 ✅
B) 1/s^2
C) e^(-sT)
D) s^2
Explanation: Triangular pulse transform.
Q87. L{periodic function f(t) with period T} = ?
A) (1/(1-e^(-sT))) ∫₀^T e^(-st)f(t)dt ✅
B) ∫₀^∞ e^(-st)f(t)dt
C) F(s-a)
D) F(s+a)
Explanation: Formula for periodic functions.
Q88. L{impulse train ∑ δ(t-nT)} = ?
A) 1/(1-e^(-sT)) ✅
B) 1/s
C) e^(-sT)
D) s
Explanation: Impulse train transform.
Q89. L{u(t)-u(t-a)} = ?
A) (1-e^(-as))/s ✅
B) 1/s
C) e^(-as)/s
D) s
Explanation: Difference of steps gives rectangular pulse.
Q90. L{δ(t)+δ(t-a)} = ?
A) 1+e^(-as) ✅
B) 1/s
C) e^(-as)
D) s
Explanation: Sum of impulses.
Q91. L{sin(at)·u(t)} = ?
A) a/(s^2+a^2) ✅
B) s/(s^2+a^2)
C) 1/(s-a)
D) 1/(s+a)
Explanation: Standard sine transform with unit step.
Q92. L{cos(at)·u(t)} = ?
A) s/(s^2+a^2) ✅
B) a/(s^2+a^2)
C) 1/(s-a)
D) 1/(s+a)
Explanation: Standard cosine transform with unit step.
Q93. L{e^(-bt)u(t)} = ?
A) 1/(s+b) ✅
B) 1/(s-b)
C) s/(s^2+b^2)
D) b/(s^2+b^2)
Explanation: Exponential decay transform.
Q94. L{t·u(t)} = ?
A) 1/s^2 ✅
B) 1/s
C) s
D) e^s
Explanation: Ramp function transform.
Q95. L{t^n·u(t)} = ?
A) n!/s^(n+1) ✅
B) 1/s^n
C) s^n/n!
D) (n+1)!/s^n
Explanation: General power function transform.
Q96. L{sin(at)/t} = ?
A) tan⁻¹(a/s) ✅
B) a/(s^2+a^2)
C) s/(s^2+a^2)
D) 1/s
Explanation: Special transform identity.
Q97. L{cos(at)/t} = ?
A) ln(s/a) ✅
B) s/(s^2+a^2)
C) a/(s^2+a^2)
D) 1/s
Explanation: Special transform identity.
Q98. L{δ'(t)} = ?
A) s ✅
B) 1
C) 0
D) 1/s
Explanation: Derivative of impulse transform.
Q99. L{δ''(t)} = ?
A) s^2 ✅
B) 1
C) 0
D) 1/s^2
Explanation: Second derivative of impulse transform.
Q100. Laplace transform is most effective for:
A) Linear time-invariant systems ✅
B) Non-linear systems
C) Random systems
D) Quantum systems
Explanation: Laplace transform applies to LTI systems.
Q101. Solve using Laplace: y'(t) + y(t) = 1, y(0)=0.
A) y(t) = 1 - e^(-t) ✅
B) y(t) = e^(-t)
C) y(t) = t
D) y(t) = sin(t)
Explanation: Transform gives (sY(s) + Y(s)) = 1/s → Y(s) = 1/(s(s+1)) → y(t)=1-e^(-t).
Q102. Solve: y''(t) - y(t) = 0, y(0)=0, y'(0)=1.
A) y(t) = sinh(t) ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Transform → (s²Y(s)-1) - Y(s)=0 → Y(s)=1/(s²-1) → y(t)=sinh(t).
Q103. Solve: y''(t)+y(t)=0, y(0)=0, y'(0)=1.
A) y(t) = sin(t) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Transform → (s²Y(s)-1)+Y(s)=0 → Y(s)=1/(s²+1) → y(t)=sin(t).
Q104. Solve: y''(t)+y(t)=0, y(0)=1, y'(0)=0.
A) y(t) = cos(t) ✅
B) y(t) = sin(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Transform → (s²Y(s)+Y(s))=s → Y(s)=s/(s²+1) → y(t)=cos(t).
Q105. Solve: y'(t)-2y(t)=0, y(0)=1.
A) y(t) = e^(2t) ✅
B) y(t) = e^(-2t)
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transform → (sY(s)-1)-2Y(s)=0 → Y(s)=1/(s-2) → y(t)=e^(2t).
Q106. Solve: y'(t)+2y(t)=0, y(0)=1.
A) y(t) = e^(-2t) ✅
B) y(t) = e^(2t)
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transform → (sY(s)-1)+2Y(s)=0 → Y(s)=1/(s+2) → y(t)=e^(-2t).
Q107. Solve: y''(t)-y'(t)-2y(t)=0, y(0)=0, y'(0)=1.
A) y(t) = (1/√2) sinh(√2 t) ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Characteristic roots → s²-s-2=0 → roots 2,-1 → solution matches sinh form.
Q108. Solve: y''(t)+4y(t)=0, y(0)=0, y'(0)=2.
A) y(t) = sin(2t) ✅
B) y(t) = cos(2t)
C) y(t) = e^(2t)
D) y(t) = sinh(2t)
Explanation: Transform → Y(s)=2/(s²+4) → y(t)=sin(2t).
Q109. Solve: y''(t)+4y(t)=0, y(0)=1, y'(0)=0.
A) y(t) = cos(2t) ✅
B) y(t) = sin(2t)
C) y(t) = e^(2t)
D) y(t) = sinh(2t)
Explanation: Transform → Y(s)=s/(s²+4) → y(t)=cos(2t).
Q110. Solve: y'(t)+y(t)=e^(-t), y(0)=0.
A) y(t) = t e^(-t) ✅
B) y(t) = e^(-t)
C) y(t) = 1-e^(-t)
D) y(t) = sin(t)
Explanation: Transform → (sY(s)) + Y(s) = 1/(s+1) → Y(s)=1/(s(s+1)²) → y(t)=t e^(-t).
Q111. Solve: y'(t)+y(t)=cos(t), y(0)=0.
A) y(t) = (1/2)(sin(t)-cos(t)+e^(-t)) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^(-t)
Explanation: Transform → Y(s)=s/(s²+1)(s+1) → partial fractions → solution.
Q112. Solve: y'(t)+y(t)=sin(t), y(0)=0.
A) y(t) = (1/2)(1-cos(t)-e^(-t)) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^(-t)
Explanation: Transform → Y(s)=1/(s²+1)(s+1) → partial fractions → solution.
Q113. Solve: y''(t)+y'(t)=0, y(0)=1, y'(0)=0.
A) y(t) = 1 ✅
B) y(t) = e^t
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Transform → (s²Y(s)-s) + (sY(s)-1)=0 → Y(s)=1/s → y(t)=1.
Q114. Solve: y''(t)-y'(t)=0, y(0)=0, y'(0)=1.
A) y(t) = t ✅
B) y(t) = e^t
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Transform → (s²Y(s)-1) - (sY(s))=0 → Y(s)=1/s² → y(t)=t.
Q115. Solve: y''(t)+y(t)=sin(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(sin(t)-t cos(t)) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Transform → Y(s)=1/(s²+1)(s²+1) → partial fractions → solution.
Q116. Solve: y''(t)+y(t)=cos(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1-cos(t)-t sin(t)) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Transform → Y(s)=s/(s²+1)(s²+1) → partial fractions → solution.
Q117. Solve: y''(t)-y(t)=e^t, y(0)=0, y'(0)=0.
A) y(t) = (1/2)(e^t - sinh(t)) ✅
B) y(t) = e^t
C) y(t) = sinh(t)
D) y(t) = cosh(t)
Explanation: Transform → Y(s)=1/(s-1)(s²-1) → partial fractions → solution.
Q118. Solve: y''(t)-y(t)=cosh(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(cosh(t)-sinh(t)) ✅
B) y(t) = cosh(t)
C) y(t) = sinh(t)
D) y(t) = e^t
Explanation: Transform → Y(s)=s/(s²-1)(s²-1) → partial fractions → solution.
Q119. Solve: y''(t)+y(t)=e^(-t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(e^(-t)-cos(t)+sin(t)) ✅
B) y(t) = e^(-t)
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Transform → Y(s)=1/(s+1)(s²+1) → partial fractions → solution.
Q120. Solve: y''(t)+y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = t - sin(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Transform → Y(s)=1/s²(s²+1) → partial fractions → y(t)=t - sin(t).
Q121. Solve: y''(t)-y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Transform → Y(s)=1/s²(s²-1) → inverse gives sinh(t)-t.
Q122. Solve: y''(t)+2y'(t)+y(t)=0, y(0)=1, y'(0)=0.
A) y(t) = e^(-t) ✅
B) y(t) = e^(t)
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Characteristic equation (s+1)²=0 → solution y(t)=e^(-t).
Q123. Solve: y''(t)+4y'(t)+4y(t)=0, y(0)=1, y'(0)=0.
A) y(t) = e^(-2t) ✅
B) y(t) = e^(2t)
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Characteristic equation (s+2)²=0 → solution y(t)=e^(-2t).
Q124. Solve: y''(t)-4y(t)=0, y(0)=0, y'(0)=2.
A) y(t) = sinh(2t) ✅
B) y(t) = cosh(2t)
C) y(t) = e^(2t)
D) y(t) = sin(2t)
Explanation: Transform → Y(s)=2/(s²-4) → inverse gives sinh(2t).
Q125. Solve: y''(t)-4y(t)=0, y(0)=1, y'(0)=0.
A) y(t) = cosh(2t) ✅
B) y(t) = sinh(2t)
C) y(t) = e^(2t)
D) y(t) = sin(2t)
Explanation: Transform → Y(s)=s/(s²-4) → inverse gives cosh(2t).
Q126. Solve: y'(t)+y(t)=t, y(0)=0.
A) y(t) = t-1+e^(-t) ✅
B) y(t) = t
C) y(t) = e^(-t)
D) y(t) = sin(t)
Explanation: Transform → Y(s)=1/s²(s+1) → inverse gives t-1+e^(-t).
Q127. Solve: y'(t)+y(t)=t², y(0)=0.
A) y(t) = t²-2t+2-2e^(-t) ✅
B) y(t) = t²
C) y(t) = e^(-t)
D) y(t) = sin(t)
Explanation: Transform → Y(s)=2/s³(s+1) → inverse gives polynomial minus exponential.
Q128. Solve: y''(t)+y(t)=cos(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1-cos(t)-t sin(t)) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Transform → Y(s)=s/(s²+1)(s²+1) → partial fractions → solution.
Q129. Solve: y''(t)+y(t)=sin(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(sin(t)-t cos(t)) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Transform → Y(s)=1/(s²+1)(s²+1) → partial fractions → solution.
Q130. Solve: y''(t)+y(t)=e^(-t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(e^(-t)-cos(t)+sin(t)) ✅
B) y(t) = e^(-t)
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Transform → Y(s)=1/(s+1)(s²+1) → inverse gives combination.
Q131. Solve: y''(t)-y(t)=e^t, y(0)=0, y'(0)=0.
A) y(t) = (1/2)(e^t - sinh(t)) ✅
B) y(t) = e^t
C) y(t) = sinh(t)
D) y(t) = cosh(t)
Explanation: Transform → Y(s)=1/(s-1)(s²-1) → inverse gives combination.
Q132. Solve: y''(t)-y(t)=cosh(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(cosh(t)-sinh(t)) ✅
B) y(t) = cosh(t)
C) y(t) = sinh(t)
D) y(t) = e^t
Explanation: Transform → Y(s)=s/(s²-1)(s²-1) → inverse gives combination.
Q133. Solve: y''(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = sin(t) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Impulse input → transfer function 1/(s²+1) → output sin(t).
Q134. Solve: y''(t)+2y'(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = e^(-t) ✅
B) y(t) = e^(t)
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Impulse response of system with (s+1)² denominator → e^(-t).
Q135. Solve: y''(t)+4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)sin(2t) ✅
B) y(t) = cos(2t)
C) y(t) = e^(2t)
D) y(t) = sinh(2t)
Explanation: Transfer function 1/(s²+4) → inverse gives (1/2)sin(2t).
Q136. Solve: y''(t)-4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)sinh(2t) ✅
B) y(t) = cosh(2t)
C) y(t) = e^(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s²-4) → inverse gives (1/2)sinh(2t).
Q137. Solve: y''(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = 1 - cos(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Step input → transfer function 1/(s²+1) → output 1-cos(t).
Q138. Solve: y''(t)+2y'(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = t e^(-t) ✅
B) y(t) = e^(-t)
C) y(t) = 1-e^(-t)
D) y(t) = sin(t)
Explanation: Step input → transfer function 1/(s+1)² → output t e^(-t).
Q139. Solve: y''(t)+4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1-cos(2t)) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^(2t)
Explanation: Step input → transfer function 1/(s²+4) → inverse gives (1/2)(1-cos(2t)).
Q140. Solve: y''(t)-4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(cosh(2t)-1) ✅
B) y(t) = sinh(2t)
C) y(t) = cosh(2t)
D) y(t) = e^(2t)
Explanation: Step input → transfer function 1/(s²-4) → inverse gives (1/4)(cosh(2t)-1).
Q141. Solve: y'(t)+y(t)=u(t), y(0)=0.
A) y(t) = 1 - e^(-t) ✅
B) y(t) = e^(-t)
C) y(t) = t
D) y(t) = sin(t)
Explanation: Step input → transfer function 1/(s+1) → inverse gives 1-e^(-t).
Q142. Solve: y'(t)-y(t)=u(t), y(0)=0.
A) y(t) = e^t - 1 ✅
B) y(t) = e^(-t)
C) y(t) = t
D) y(t) = sin(t)
Explanation: Step input → transfer function 1/(s-1) → inverse gives e^t-1.
Q143. Solve: y''(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = sin(t) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Impulse input → transfer function 1/(s²+1) → inverse gives sin(t).
Q144. Solve: y''(t)+2y'(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = e^(-t) ✅
B) y(t) = e^(t)
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Impulse response of system with (s+1)² denominator → e^(-t).
Q145. Solve: y''(t)+4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)sin(2t) ✅
B) y(t) = cos(2t)
C) y(t) = e^(2t)
D) y(t) = sinh(2t)
Explanation: Transfer function 1/(s²+4) → inverse gives (1/2)sin(2t).
Q146. Solve: y''(t)-4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)sinh(2t) ✅
B) y(t) = cosh(2t)
C) y(t) = e^(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s²-4) → inverse gives (1/2)sinh(2t).
Q147. Solve: y'(t)+2y(t)=δ(t), y(0)=0.
A) y(t) = e^(-2t) ✅
B) y(t) = e^(2t)
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s+2) → inverse gives e^(-2t).
Q148. Solve: y'(t)-2y(t)=δ(t), y(0)=0.
A) y(t) = e^(2t) ✅
B) y(t) = e^(-2t)
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s-2) → inverse gives e^(2t).
Q149. Solve: y''(t)+y(t)=cos(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1-cos(t)-t sin(t)) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Transform → partial fractions → solution.
Q150. Solve: y''(t)+y(t)=sin(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(sin(t)-t cos(t)) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Transform → partial fractions → solution.
Q151. Solve: y''(t)+y(t)=e^(-t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(e^(-t)-cos(t)+sin(t)) ✅
B) y(t) = e^(-t)
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Transform → Y(s)=1/(s+1)(s²+1) → inverse gives combination.
Q152. Solve: y''(t)-y(t)=e^t, y(0)=0, y'(0)=0.
A) y(t) = (1/2)(e^t - sinh(t)) ✅
B) y(t) = e^t
C) y(t) = sinh(t)
D) y(t) = cosh(t)
Explanation: Transform → Y(s)=1/(s-1)(s²-1) → inverse gives combination.
Q153. Solve: y''(t)-y(t)=cosh(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(cosh(t)-sinh(t)) ✅
B) y(t) = cosh(t)
C) y(t) = sinh(t)
D) y(t) = e^t
Explanation: Transform → Y(s)=s/(s²-1)(s²-1) → inverse gives combination.
Q154. Solve: y''(t)+y(t)=δ(t-π), y(0)=0, y'(0)=0.
A) y(t) = sin(t-Ï€)u(t-Ï€) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Shifted impulse → solution sin(t-Ï€)u(t-Ï€).
Q155. Solve: y''(t)+y(t)=u(t-Ï€), y(0)=0, y'(0)=0.
A) y(t) = (1-cos(t-Ï€))u(t-Ï€) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Shifted step → solution (1-cos(t-Ï€))u(t-Ï€).
Q156. Solve: y'(t)+y(t)=δ(t-1), y(0)=0.
A) y(t) = e^(-(t-1))u(t-1) ✅
B) y(t) = e^(-t)
C) y(t) = e^(t)
D) y(t) = sin(t)
Explanation: Shifted impulse → solution e^(-(t-1))u(t-1).
Q157. Solve: y'(t)+y(t)=u(t-1), y(0)=0.
A) y(t) = (1-e^(-(t-1)))u(t-1) ✅
B) y(t) = e^(-t)
C) y(t) = e^(t)
D) y(t) = sin(t)
Explanation: Shifted step → solution (1-e^(-(t-1)))u(t-1).
Q158. Solve: y''(t)+4y(t)=δ(t-π), y(0)=0, y'(0)=0.
A) y(t) = (1/2)sin(2(t-Ï€))u(t-Ï€) ✅
B) y(t) = cos(2t)
C) y(t) = e^(2t)
D) y(t) = sinh(2t)
Explanation: Shifted impulse → solution (1/2)sin(2(t-Ï€))u(t-Ï€).
Q159. Solve: y''(t)+4y(t)=u(t-Ï€), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1-cos(2(t-Ï€)))u(t-Ï€) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^(2t)
Explanation: Step input shifted → transfer function 1/(s²+4) → inverse gives (1/2)(1-cos(2(t-Ï€)))u(t-Ï€).
Q160. Solve: y''(t)-4y(t)=u(t-Ï€), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(cosh(2(t-Ï€))-1)u(t-Ï€) ✅
B) y(t) = sinh(2t)
C) y(t) = cosh(2t)
D) y(t) = e^(2t)
Explanation: Step input shifted → transfer function 1/(s²-4) → inverse gives (1/4)(cosh(2(t-Ï€))-1)u(t-Ï€).
Q161. Solve: y'(t)+y(t)=δ(t-π), y(0)=0.
A) y(t) = e^(-(t-Ï€))u(t-Ï€) ✅
B) y(t) = e^(-t)
C) y(t) = e^(t)
D) y(t) = sin(t)
Explanation: Shifted impulse → solution e^(-(t-Ï€))u(t-Ï€).
Q162. Solve: y'(t)+y(t)=u(t-Ï€), y(0)=0.
A) y(t) = (1-e^(-(t-Ï€)))u(t-Ï€) ✅
B) y(t) = e^(-t)
C) y(t) = e^(t)
D) y(t) = sin(t)
Explanation: Shifted step → solution (1-e^(-(t-Ï€)))u(t-Ï€).
Q163. Solve: y''(t)+y(t)=δ(t-a), y(0)=0, y'(0)=0.
A) y(t) = sin(t-a)u(t-a) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Impulse shifted → solution sin(t-a)u(t-a).
Q164. Solve: y''(t)+y(t)=u(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1-cos(t-a))u(t-a) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Step shifted → solution (1-cos(t-a))u(t-a).
Q165. Solve: y'(t)+2y(t)=u(t-a), y(0)=0.
A) y(t) = (1-e^(-2(t-a)))u(t-a) ✅
B) y(t) = e^(-2t)
C) y(t) = e^(2t)
D) y(t) = sin(t)
Explanation: Step shifted → solution (1-e^(-2(t-a)))u(t-a).
Q166. Solve: y'(t)-2y(t)=u(t-a), y(0)=0.
A) y(t) = (e^(2(t-a))-1)u(t-a) ✅
B) y(t) = e^(-2t)
C) y(t) = e^(2t)
D) y(t) = sin(t)
Explanation: Step shifted → solution (e^(2(t-a))-1)u(t-a).
Q167. Solve: y''(t)+4y(t)=δ(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/2)sin(2(t-a))u(t-a) ✅
B) y(t) = cos(2t)
C) y(t) = e^(2t)
D) y(t) = sinh(2t)
Explanation: Impulse shifted → solution (1/2)sin(2(t-a))u(t-a).
Q168. Solve: y''(t)-4y(t)=δ(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/2)sinh(2(t-a))u(t-a) ✅
B) y(t) = cosh(2t)
C) y(t) = e^(2t)
D) y(t) = sin(2t)
Explanation: Impulse shifted → solution (1/2)sinh(2(t-a))u(t-a).
Q169. Solve: y'(t)+y(t)=cos(t), y(0)=0.
A) y(t) = (1/2)(sin(t)-cos(t)+e^(-t)) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^(-t)
Explanation: Transform → Y(s)=s/(s²+1)(s+1) → partial fractions → solution.
Q170. Solve: y'(t)+y(t)=sin(t), y(0)=0.
A) y(t) = (1/2)(1-cos(t)-e^(-t)) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^(-t)
Explanation: Transform → Y(s)=1/(s²+1)(s+1) → partial fractions → solution.
Q171. Solve: y''(t)+y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = t - sin(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Transform → Y(s)=1/s²(s²+1) → inverse gives t-sin(t).
Q172. Solve: y''(t)-y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Transform → Y(s)=1/s²(s²-1) → inverse gives sinh(t)-t.
Q173. Solve: y''(t)+2y'(t)+y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = t e^(-t) ✅
B) y(t) = e^(-t)
C) y(t) = 1-e^(-t)
D) y(t) = sin(t)
Explanation: Transfer function 1/(s+1)² → inverse gives t e^(-t).
Q174. Solve: y''(t)+4y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = (1/2)(t - sin(2t)) ✅
B) y(t) = cos(2t)
C) y(t) = sin(2t)
D) y(t) = e^(2t)
Explanation: Transform → Y(s)=1/s²(s²+4) → inverse gives (1/2)(t-sin(2t)).
Q175. Solve: y''(t)-4y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = (1/4)(sinh(2t)-2t) ✅
B) y(t) = cosh(2t)
C) y(t) = sinh(2t)
D) y(t) = e^(2t)
Explanation: Transform → Y(s)=1/s²(s²-4) → inverse gives (1/4)(sinh(2t)-2t).
Q176. Solve: y'(t)+y(t)=t, y(0)=0.
A) y(t) = t-1+e^(-t) ✅
B) y(t) = t
C) y(t) = e^(-t)
D) y(t) = sin(t)
Explanation: Transform → Y(s)=1/s²(s+1) → inverse gives t-1+e^(-t).
Q177. Solve: y'(t)+y(t)=t², y(0)=0.
A) y(t) = t²-2t+2-2e^(-t) ✅
B) y(t) = t²
C) y(t) = e^(-t)
D) y(t) = sin(t)
Explanation: Transform → Y(s)=2/s³(s+1) → inverse gives polynomial minus exponential.
Q178. Solve: y''(t)+y(t)=t², y(0)=0, y'(0)=0.
A) y(t) = t² - 2 + 2cos(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Transform → Y(s)=2/s³(s²+1) → inverse gives t²-2+2cos(t).
Q179. Solve: y''(t)-y(t)=t², y(0)=0, y'(0)=0.
A) y(t) = (1/2)(sinh(t)-t²) ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Transform → Y(s)=2/s³(s²-1) → inverse gives (1/2)(sinh(t)-t²).
Q180. Solve: y'(t)+y(t)=t², y(0)=0.
A) y(t) = t²-2t+2-2e^(-t) ✅
B) y(t) = t²
C) y(t) = e^(-t)
D) y(t) = sin(t)
Explanation: Transform → Y(s)=2/s³(s+1) → inverse gives polynomial minus exponential.
Q181. Solve: y''(t)+y(t)=t³, y(0)=0, y'(0)=0.
A) y(t) = t³ - 6t + 6sin(t) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Transform → Y(s)=6/s⁴(s²+1) → inverse gives t³-6t+6sin(t).
Q182. Solve: y''(t)-y(t)=t³, y(0)=0, y'(0)=0.
A) y(t) = (1/2)(sinh(t)-t³) ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Transform → Y(s)=6/s⁴(s²-1) → inverse gives (1/2)(sinh(t)-t³).
Q183. Solve: y'(t)+2y(t)=t³, y(0)=0.
A) y(t) = t³ - 3t² + 6t - 6 + 6e^(-2t) ✅
B) y(t) = t³
C) y(t) = e^(-2t)
D) y(t) = sin(t)
Explanation: Transform → Y(s)=6/s⁴(s+2) → inverse gives polynomial minus exponential.
Q184. Solve: y''(t)+y(t)=t⁴, y(0)=0, y'(0)=0.
A) y(t) = t⁴ - 12t² + 24 - 24cos(t) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Transform → Y(s)=24/s⁵(s²+1) → inverse gives polynomial with cosine term.
Q185. Solve: y''(t)-y(t)=t⁴, y(0)=0, y'(0)=0.
A) y(t) = (1/2)(sinh(t)-t⁴) ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Transform → Y(s)=24/s⁵(s²-1) → inverse gives (1/2)(sinh(t)-t⁴).
Q186. Solve: y'(t)+y(t)=t⁴, y(0)=0.
A) y(t) = t⁴ - 4t³ + 12t² - 24t + 24 - 24e^(-t) ✅
B) y(t) = t⁴
C) y(t) = e^(-t)
D) y(t) = sin(t)
Explanation: Transform → Y(s)=24/s⁵(s+1) → inverse gives polynomial minus exponential.
Q187. Solve: y''(t)+y(t)=e^(-t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(e^(-t)-cos(t)+sin(t)) ✅
B) y(t) = e^(-t)
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Transform → Y(s)=1/(s+1)(s²+1) → inverse gives combination.
Q188. Solve: y''(t)-y(t)=e^t, y(0)=0, y'(0)=0.
A) y(t) = (1/2)(e^t - sinh(t)) ✅
B) y(t) = e^t
C) y(t) = sinh(t)
D) y(t) = cosh(t)
Explanation: Transform → Y(s)=1/(s-1)(s²-1) → inverse gives combination.
Q189. Solve: y''(t)-y(t)=cosh(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(cosh(t)-sinh(t)) ✅
B) y(t) = cosh(t)
C) y(t) = sinh(t)
D) y(t) = e^t
Explanation: Transform → Y(s)=s/(s²-1)(s²-1) → inverse gives combination.
Q190. Solve: y''(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = sin(t) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Impulse input → transfer function 1/(s²+1) → output sin(t).
Q191. Solve: y''(t)+2y'(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = e^(-t) ✅
B) y(t) = e^(t)
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Impulse response of system with (s+1)² denominator → e^(-t).
Q192. Solve: y''(t)+4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)sin(2t) ✅
B) y(t) = cos(2t)
C) y(t) = e^(2t)
D) y(t) = sinh(2t)
Explanation: Transfer function 1/(s²+4) → inverse gives (1/2)sin(2t).
Q193. Solve: y''(t)-4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)sinh(2t) ✅
B) y(t) = cosh(2t)
C) y(t) = e^(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s²-4) → inverse gives (1/2)sinh(2t).
Q194. Solve: y'(t)+2y(t)=δ(t), y(0)=0.
A) y(t) = e^(-2t) ✅
B) y(t) = e^(2t)
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s+2) → inverse gives e^(-2t).
Q195. Solve: y'(t)-2y(t)=δ(t), y(0)=0.
A) y(t) = e^(2t) ✅
B) y(t) = e^(-2t)
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s-2) → inverse gives e^(2t).
Q196. Solve: y''(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = 1 - cos(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Step input → transfer function 1/(s²+1) → output 1-cos(t).
Q197. Solve: y''(t)+2y'(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = t e^(-t) ✅
B) y(t) = e^(-t)
C) y(t) = 1-e^(-t)
D) y(t) = sin(t)
Explanation: Step input → transfer function 1/(s+1)² → inverse gives t e^(-t).
Q198. Solve: y''(t)+4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1-cos(2t)) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^(2t)
Explanation: Step input → transfer function 1/(s²+4) → inverse gives (1/2)(1-cos(2t)).
Q199. Solve: y''(t)-4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(cosh(2t)-1) ✅
B) y(t) = sinh(2t)
C) y(t) = cosh(2t)
D) y(t) = e^(2t)
Explanation: Step input → transfer function 1/(s²-4) → inverse gives (1/4)(cosh(2t)-1).
Q200. Solve: y'(t)+y(t)=u(t), y(0)=0.
A) y(t) = 1 - e^(-t) ✅
B) y(t) = e^(-t)
C) y(t) = t
D) y(t) = sin(t)
Explanation: Step input → transfer function 1/(s+1) → inverse gives 1-e^(-t).
Q201. Solve: y'(t)-y(t)=u(t), y(0)=0.
A) y(t) = e^t - 1 ✅
B) y(t) = e^(-t)
C) y(t) = t
D) y(t) = sin(t)
Explanation: Step input → transfer function 1/(s-1) → inverse gives e^t-1.
Q202. Solve: y''(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = sin(t) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Impulse input → transfer function 1/(s²+1) → output sin(t).
Q203. Solve: y''(t)+2y'(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = e^(-t) ✅
B) y(t) = e^(t)
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Impulse response of system with (s+1)² denominator → e^(-t).
Q204. Solve: y''(t)+4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)sin(2t) ✅
B) y(t) = cos(2t)
C) y(t) = e^(2t)
D) y(t) = sinh(2t)
Explanation: Transfer function 1/(s²+4) → inverse gives (1/2)sin(2t).
Q205. Solve: y''(t)-4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)sinh(2t) ✅
B) y(t) = cosh(2t)
C) y(t) = e^(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s²-4) → inverse gives (1/2)sinh(2t).
Q206. Solve: y'(t)+2y(t)=δ(t), y(0)=0.
A) y(t) = e^(-2t) ✅
B) y(t) = e^(2t)
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s+2) → inverse gives e^(-2t).
Q207. Solve: y'(t)-2y(t)=δ(t), y(0)=0.
A) y(t) = e^(2t) ✅
B) y(t) = e^(-2t)
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s-2) → inverse gives e^(2t).
Q208. Solve: y''(t)+y(t)=u(t-Ï€), y(0)=0, y'(0)=0.
A) y(t) = (1-cos(t-Ï€))u(t-Ï€) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Step shifted → solution (1-cos(t-Ï€))u(t-Ï€).
Q209. Solve: y''(t)+y(t)=δ(t-π), y(0)=0, y'(0)=0.
A) y(t) = sin(t-Ï€)u(t-Ï€) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Impulse shifted → solution sin(t-Ï€)u(t-Ï€).
Q210. Solve: y'(t)+y(t)=δ(t-π), y(0)=0.
A) y(t) = e^(-(t-Ï€))u(t-Ï€) ✅
B) y(t) = e^(-t)
C) y(t) = e^(t)
D) y(t) = sin(t)
Explanation: Shifted impulse → solution e^(-(t-Ï€))u(t-Ï€).
Q211. Solve: y'(t)+y(t)=u(t-Ï€), y(0)=0.
A) y(t) = (1-e^(-(t-Ï€)))u(t-Ï€) ✅
B) y(t) = e^(-t)
C) y(t) = e^(t)
D) y(t) = sin(t)
Explanation: Shifted step → solution (1-e^(-(t-Ï€)))u(t-Ï€).
Q212. Solve: y''(t)+4y(t)=δ(t-π), y(0)=0, y'(0)=0.
A) y(t) = (1/2)sin(2(t-Ï€))u(t-Ï€) ✅
B) y(t) = cos(2t)
C) y(t) = e^(2t)
D) y(t) = sinh(2t)
Explanation: Shifted impulse → solution (1/2)sin(2(t-Ï€))u(t-Ï€).
Q213. Solve: y''(t)+4y(t)=u(t-Ï€), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1-cos(2(t-Ï€)))u(t-Ï€) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^(2t)
Explanation: Shifted step → solution (1/2)(1-cos(2(t-Ï€)))u(t-Ï€).
Q214. Solve: y''(t)-4y(t)=δ(t-π), y(0)=0, y'(0)=0.
A) y(t) = (1/2)sinh(2(t-Ï€))u(t-Ï€) ✅
B) y(t) = cosh(2t)
C) y(t) = e^(2t)
D) y(t) = sin(2t)
Explanation: Shifted impulse → solution (1/2)sinh(2(t-Ï€))u(t-Ï€).
Q215. Solve: y''(t)-4y(t)=u(t-Ï€), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(cosh(2(t-Ï€))-1)u(t-Ï€) ✅
B) y(t) = sinh(2t)
C) y(t) = cosh(2t)
D) y(t) = e^(2t)
Explanation: Shifted step → solution (1/4)(cosh(2(t-Ï€))-1)u(t-Ï€).
Q216. Solve: y'(t)+2y(t)=u(t-Ï€), y(0)=0.
A) y(t) = (1 - e^(-2(t-Ï€))) u(t-Ï€) ✅
B) y(t) = e^(-2t)
C) y(t) = e^(2t)
D) y(t) = sin(2t)
Explanation: Shifted step input; transfer function 1/(s+2); time-shift → (1 - e^(-2(t-Ï€)))u(t-Ï€).
Q217. Solve: y'(t)-2y(t)=u(t-Ï€), y(0)=0.
A) y(t) = (e^(2(t-Ï€)) - 1) u(t-Ï€) ✅
B) y(t) = e^(-2t)
C) y(t) = e^(2t)
D) y(t) = sin(2t)
Explanation: Shifted step input; transfer function 1/(s-2); time-shift → (e^(2(t-Ï€)) - 1)u(t-Ï€).
Q218. Solve: y''(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = 1 - cos(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Step input; transfer function 1/(s²+1); inverse → 1 - cos(t).
Q219. Solve: y''(t)+2y'(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = t e^(-t) ✅
B) y(t) = e^(-t)
C) y(t) = 1 - e^(-t)
D) y(t) = sin(t)
Explanation: Step input; transfer function 1/(s+1)²; inverse → t e^(-t).
Q220. Solve: y''(t)+4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1 - cos(2t)) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^(2t)
Explanation: Step input; transfer function 1/(s²+4); inverse → (1/2)(1 - cos(2t)).
Q221. Solve: y''(t)-4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(cosh(2t) - 1) ✅
B) y(t) = sinh(2t)
C) y(t) = cosh(2t)
D) y(t) = e^(2t)
Explanation: Step input; transfer function 1/(s²-4); inverse → (1/4)(cosh(2t) - 1).
Q222. Solve: y'(t)+y(t)=u(t-a), y(0)=0.
A) y(t) = (1 - e^{-(t-a)}) u(t-a) ✅
B) y(t) = e^{-t}
C) y(t) = e^{t}
D) y(t) = sin(t)
Explanation: Shifted step; transfer function 1/(s+1); time-shift → (1 - e^{-(t-a)})u(t-a).
Q223. Solve: y'(t)-y(t)=u(t-a), y(0)=0.
A) y(t) = (e^{(t-a)} - 1) u(t-a) ✅
B) y(t) = e^{-t}
C) y(t) = e^{t}
D) y(t) = sin(t)
Explanation: Shifted step; transfer function 1/(s-1); time-shift → (e^{(t-a)} - 1)u(t-a).
Q224. Solve: y''(t)+y(t)=δ(t-a), y(0)=0, y'(0)=0.
A) y(t) = sin(t-a) u(t-a) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Shifted impulse; transfer function 1/(s²+1); inverse → sin(t-a)u(t-a).
Q225. Solve: y''(t)+y(t)=u(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1 - cos(t-a)) u(t-a) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Shifted step; transfer function 1/(s²+1); inverse → (1 - cos(t-a))u(t-a).
Q226. Solve: y''(t)+4y(t)=δ(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sin(2(t-a)) u(t-a) ✅
B) y(t) = cos(2t)
C) y(t) = e^{2t}
D) y(t) = sinh(2t)
Explanation: Shifted impulse; transfer function 1/(s²+4); inverse → (1/2)sin(2(t-a))u(t-a).
Q227. Solve: y''(t)+4y(t)=u(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1 - cos(2(t-a))) u(t-a) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^{2t}
Explanation: Shifted step; transfer function 1/(s²+4); inverse → (1/2)(1 - cos(2(t-a)))u(t-a).
Q228. Solve: y''(t)-4y(t)=δ(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sinh(2(t-a)) u(t-a) ✅
B) y(t) = cosh(2t)
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted impulse; transfer function 1/(s²-4); inverse → (1/2)sinh(2(t-a))u(t-a).
Q229. Solve: y''(t)-4y(t)=u(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(cosh(2(t-a)) - 1) u(t-a) ✅
B) y(t) = sinh(2t)
C) y(t) = cosh(2t)
D) y(t) = e^{2t}
Explanation: Shifted step; transfer function 1/(s²-4); inverse → (1/4)(cosh(2(t-a)) - 1)u(t-a).
Q230. Solve: y'(t)+2y(t)=δ(t-a), y(0)=0.
A) y(t) = e^{-2(t-a)} u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted impulse; transfer function 1/(s+2); inverse → e^{-2(t-a)}u(t-a).
Q231. Solve: y'(t)-2y(t)=δ(t-a), y(0)=0.
A) y(t) = e^{2(t-a)} u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted impulse; transfer function 1/(s-2); inverse → e^{2(t-a)}u(t-a).
Q232. Solve: y'(t)+y(t)=cos(t), y(0)=0.
A) y(t) = (1/2)(sin(t) - cos(t) + e^{-t}) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^{-t}
Explanation: Transform → Y(s)=s/[(s²+1)(s+1)] → partial fractions → solution.
Q233. Solve: y'(t)+y(t)=sin(t), y(0)=0.
A) y(t) = (1/2)(1 - cos(t) - e^{-t}) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^{-t}
Explanation: Transform → Y(s)=1/[(s²+1)(s+1)] → partial fractions → solution.
Q234. Solve: y''(t)+y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = t - sin(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Transform → Y(s)=1/[s²(s²+1)] → inverse → t - sin(t).
Q235. Solve: y''(t)-y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Transform → Y(s)=1/[s²(s²-1)] → inverse → sinh(t) - t.
Q236. Solve: y''(t)+y(t)=t², y(0)=0, y'(0)=0.
A) y(t) = t² - 2 + 2cos(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Transform → Y(s)=2/[s³(s²+1)] → inverse gives t² - 2 + 2cos(t).
Q237. Solve: y''(t)-y(t)=t², y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t² ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Transform → Y(s)=2/[s³(s²-1)] → inverse gives sinh(t)-t².
Q238. Solve: y'(t)+y(t)=t², y(0)=0.
A) y(t) = t² - 2t + 2 - 2e^(-t) ✅
B) y(t) = t²
C) y(t) = e^(-t)
D) y(t) = sin(t)
Explanation: Transform → Y(s)=2/[s³(s+1)] → inverse gives polynomial minus exponential.
Q239. Solve: y''(t)+y(t)=t³, y(0)=0, y'(0)=0.
A) y(t) = t³ - 6t + 6sin(t) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Transform → Y(s)=6/[s⁴(s²+1)] → inverse gives t³ - 6t + 6sin(t).
Q240. Solve: y''(t)-y(t)=t³, y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t³ ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Transform → Y(s)=6/[s⁴(s²-1)] → inverse gives sinh(t)-t³.
Q241. Solve: y'(t)+2y(t)=t³, y(0)=0.
A) y(t) = t³ - 3t² + 6t - 6 + 6e^(-2t) ✅
B) y(t) = t³
C) y(t) = e^(-2t)
D) y(t) = sin(t)
Explanation: Transform → Y(s)=6/[s⁴(s+2)] → inverse gives polynomial minus exponential.
Q242. Solve: y''(t)+y(t)=t⁴, y(0)=0, y'(0)=0.
A) y(t) = t⁴ - 12t² + 24 - 24cos(t) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Transform → Y(s)=24/[s⁵(s²+1)] → inverse gives polynomial with cosine term.
Q243. Solve: y''(t)-y(t)=t⁴, y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t⁴ ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Transform → Y(s)=24/[s⁵(s²-1)] → inverse gives sinh(t)-t⁴.
Q244. Solve: y'(t)+y(t)=t⁴, y(0)=0.
A) y(t) = t⁴ - 4t³ + 12t² - 24t + 24 - 24e^(-t) ✅
B) y(t) = t⁴
C) y(t) = e^(-t)
D) y(t) = sin(t)
Explanation: Transform → Y(s)=24/[s⁵(s+1)] → inverse gives polynomial minus exponential.
Q245. Solve: y''(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = sin(t) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Impulse input → transfer function 1/(s²+1) → output sin(t).
Q246. Solve: y''(t)+2y'(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = e^(-t) ✅
B) y(t) = e^(t)
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Impulse response of system with (s+1)² denominator → e^(-t).
Q247. Solve: y''(t)+4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)sin(2t) ✅
B) y(t) = cos(2t)
C) y(t) = e^(2t)
D) y(t) = sinh(2t)
Explanation: Transfer function 1/(s²+4) → inverse gives (1/2)sin(2t).
Q248. Solve: y''(t)-4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)sinh(2t) ✅
B) y(t) = cosh(2t)
C) y(t) = e^(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s²-4) → inverse gives (1/2)sinh(2t).
Q249. Solve: y'(t)+2y(t)=δ(t), y(0)=0.
A) y(t) = e^(-2t) ✅
B) y(t) = e^(2t)
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s+2) → inverse gives e^(-2t).
Q250. Solve: y'(t)-2y(t)=δ(t), y(0)=0.
A) y(t) = e^(2t) ✅
B) y(t) = e^(-2t)
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s-2) → inverse gives e^(2t).
Q251. Solve: y''(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = 1 - cos(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Step input → transfer function 1/(s²+1) → output 1-cos(t).
Q252. Solve: y''(t)+2y'(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = t e^(-t) ✅
B) y(t) = e^(-t)
C) y(t) = 1 - e^(-t)
D) y(t) = sin(t)
Explanation: Step input → transfer function 1/(s+1)² → inverse gives t e^(-t).
Q253. Solve: y''(t)+4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1 - cos(2t)) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^(2t)
Explanation: Step input → transfer function 1/(s²+4) → inverse gives (1/2)(1-cos(2t)).
Q254. Solve: y''(t)-4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(cosh(2t) - 1) ✅
B) y(t) = sinh(2t)
C) y(t) = cosh(2t)
D) y(t) = e^(2t)
Explanation: Step input → transfer function 1/(s²-4) → inverse gives (1/4)(cosh(2t)-1).
Q255. Solve: y'(t)+y(t)=u(t), y(0)=0.
A) y(t) = 1 - e^{-t} ✅
B) y(t) = e^{-t}
C) y(t) = t
D) y(t) = \sin t
Explanation: Step input; transfer function 1/(s+1); inverse → 1 - e^{-t}.
Q256. Solve: y'(t)-y(t)=u(t), y(0)=0.
A) y(t) = e^{t} - 1 ✅
B) y(t) = e^{-t}
C) y(t) = t
D) y(t) = \sin t
Explanation: Step input; transfer function 1/(s-1); inverse → e^{t} - 1.
Q257. Solve: y''(t)+y(t)=\cos t, y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{2}(1 - \cos t - t \sin t) ✅
B) y(t) = \cos t
C) y(t) = \sin t
D) y(t) = e^{t}
Explanation: Y(s)=s/[(s^{2}+1)(s^{2}+1)]; partial fractions → stated form.
Q258. Solve: y''(t)+y(t)=\sin t, y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{2}(\sin t - t \cos t) ✅
B) y(t) = \sin t
C) y(t) = \cos t
D) y(t) = e^{t}
Explanation: Y(s)=1/[(s^{2}+1)(s^{2}+1)]; partial fractions → stated form.
Q259. Solve: y''(t)+4y(t)=\cos 2t, y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{4}(1 - \cos 2t - t \sin 2t) ✅
B) y(t) = \cos 2t
C) y(t) = \sin 2t
D) y(t) = e^{2t}
Explanation: Resonant-like structure with \omega=2; partial fractions → stated form.
Q260. Solve: y''(t)+4y(t)=\sin 2t, y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{4}(\sin 2t - 2t \cos 2t) ✅
B) y(t) = \sin 2t
C) y(t) = \cos 2t
D) y(t) = e^{2t}
Explanation: Y(s)=2/[(s^{2}+4)(s^{2}+4)]; partial fractions → stated form.
Q261. Solve: y'(t)+2y(t)=\cos t, y(0)=0.
A) y(t) = \tfrac{1}{5}(2\cos t + \sin t - 2e^{-2t}) ✅
B) y(t) = \cos t
C) y(t) = \sin t
D) y(t) = e^{-2t}
Explanation: Y(s)=s/[(s^{2}+1)(s+2)]; partial fractions → stated form.
Q262. Solve: y'(t)+2y(t)=\sin t, y(0)=0.
A) y(t) = \tfrac{1}{5}(2\sin t - \cos t - e^{-2t}) ✅
B) y(t) = \sin t
C) y(t) = \cos t
D) y(t) = e^{-2t}
Explanation: Y(s)=1/[(s^{2}+1)(s+2)]; partial fractions → stated form.
Q263. Solve: y''(t)-y(t)=\cosh t, y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{2}(\cosh t - \sinh t) ✅
B) y(t) = \cosh t
C) y(t) = \sinh t
D) y(t) = e^{t}
Explanation: Y(s)=s/[(s^{2}-1)(s^{2}-1)]; partial fractions → stated form.
Q264. Solve: y''(t)-y(t)=\sinh t, y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{2}(\sinh t - t) ✅
B) y(t) = \sinh t
C) y(t) = \cosh t
D) y(t) = e^{t}
Explanation: Y(s)=1/[(s^{2}-1)(s^{2}-1)]; inverse → stated form.
Q265. Solve: y''(t)+y(t)=e^{-t}, y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{2}(e^{-t} - \cos t + \sin t) ✅
B) y(t) = e^{-t}
C) y(t) = \cos t
D) y(t) = \sin t
Explanation: Y(s)=1/[(s+1)(s^{2}+1)]; partial fractions → stated form.
Q266. Solve: y'(t)+y(t)=e^{-t}, y(0)=0.
A) y(t) = t e^{-t} ✅
B) y(t) = e^{-t}
C) y(t) = 1 - e^{-t}
D) y(t) = \sin t
Explanation: Y(s)=1/[(s+1)^{2}]; inverse → t e^{-t}.
Q267. Solve: y''(t)+2y'(t)+y(t)=e^{-t}, y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{2} t^{2} e^{-t} ✅
B) y(t) = t e^{-t}
C) y(t) = e^{-t}
D) y(t) = \sin t
Explanation: Repeated pole at s=-1; Y(s)=1/(s+1)^{3}; inverse → (t^{2}/2)e^{-t}.
Q268. Solve: y''(t)+y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = t - \sin t ✅
B) y(t) = \sin t
C) y(t) = \cos t
D) y(t) = e^{t}
Explanation: Y(s)=1/[s^{2}(s^{2}+1)]; inverse → t - \sin t.
Q269. Solve: y'(t)+y(t)=t, y(0)=0.
A) y(t) = t - 1 + e^{-t} ✅
B) y(t) = t
C) y(t) = e^{-t}
D) y(t) = \sin t
Explanation: Y(s)=1/[s^{2}(s+1)]; inverse → t - 1 + e^{-t}.
Q270. Solve: y''(t)-4y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{4}(\sinh 2t - 2t) ✅
B) y(t) = \cosh 2t
C) y(t) = \sinh 2t
D) y(t) = e^{2t}
Explanation: Y(s)=1/[s^{2}(s^{2}-4)]; inverse → (1/4)(\sinh 2t - 2t).
Q271. Solve: y''(t)+4y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{2}(t - \sin 2t) ✅
B) y(t) = \cos 2t
C) y(t) = \sin 2t
D) y(t) = e^{2t}
Explanation: Y(s)=1/[s^{2}(s^{2}+4)]; inverse → (1/2)(t - \sin 2t).
Q272. Solve: y'(t)+2y(t)=t, y(0)=0.
A) y(t) = t - \tfrac{1}{2} + \tfrac{1}{2} e^{-2t} ✅
B) y(t) = t
C) y(t) = e^{-2t}
D) y(t) = \sin t
Explanation: Y(s)=1/[s^{2}(s+2)]; partial fractions → stated form.
Q273. Solve: y'(t)+2y(t)=t^{2}, y(0)=0.
A) y(t) = t^{2} - t + \tfrac{1}{2} - \tfrac{1}{2} e^{-2t} ✅
B) y(t) = t^{2}
C) y(t) = e^{-2t}
D) y(t) = \sin t
Explanation: Y(s)=2/[s^{3}(s+2)]; partial fractions → stated form.
Q274. Solve: y''(t)+y(t)=t^{2}, y(0)=0, y'(0)=0.
A) y(t) = t^{2} - 2 + 2\cos t ✅
B) y(t) = \sin t
C) y(t) = \cos t
D) y(t) = e^{t}
Explanation: Y(s)=2/[s^{3}(s^{2}+1)]; inverse → t^{2} - 2 + 2\cos t.
Q275. Solve: y''(t)-y(t)=t^{2}, y(0)=0, y'(0)=0.
A) y(t) = \sinh t - t^{2} ✅
B) y(t) = \cosh t
C) y(t) = e^{t}
D) y(t) = \sin t
Explanation: Y(s)=2/[s^{3}(s^{2}-1)]; inverse → \sinh t - t^{2}.
Q276. Solve: y''(t)+y(t)=t³, y(0)=0, y'(0)=0.
A) y(t) = t³ - 6t + 6sin(t) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Y(s)=6/[s⁴(s²+1)] → inverse gives t³ - 6t + 6sin(t).
Q277. Solve: y''(t)-y(t)=t³, y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t³ ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Y(s)=6/[s⁴(s²-1)] → inverse gives sinh(t)-t³.
Q278. Solve: y'(t)+2y(t)=t³, y(0)=0.
A) y(t) = t³ - 3t² + 6t - 6 + 6e^(-2t) ✅
B) y(t) = t³
C) y(t) = e^(-2t)
D) y(t) = sin(t)
Explanation: Y(s)=6/[s⁴(s+2)] → inverse gives polynomial minus exponential.
Q279. Solve: y''(t)+y(t)=t⁴, y(0)=0, y'(0)=0.
A) y(t) = t⁴ - 12t² + 24 - 24cos(t) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Y(s)=24/[s⁵(s²+1)] → inverse gives polynomial with cosine term.
Q280. Solve: y''(t)-y(t)=t⁴, y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t⁴ ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Y(s)=24/[s⁵(s²-1)] → inverse gives sinh(t)-t⁴.
Q281. Solve: y'(t)+y(t)=t⁴, y(0)=0.
A) y(t) = t⁴ - 4t³ + 12t² - 24t + 24 - 24e^(-t) ✅
B) y(t) = t⁴
C) y(t) = e^(-t)
D) y(t) = sin(t)
Explanation: Y(s)=24/[s⁵(s+1)] → inverse gives polynomial minus exponential.
Q282. Solve: y''(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = sin(t) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Impulse input → transfer function 1/(s²+1) → output sin(t).
Q283. Solve: y''(t)+2y'(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = e^(-t) ✅
B) y(t) = e^(t)
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Impulse response of system with (s+1)² denominator → e^(-t).
Q284. Solve: y''(t)+4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)sin(2t) ✅
B) y(t) = cos(2t)
C) y(t) = e^(2t)
D) y(t) = sinh(2t)
Explanation: Transfer function 1/(s²+4) → inverse gives (1/2)sin(2t).
Q285. Solve: y''(t)-4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)sinh(2t) ✅
B) y(t) = cosh(2t)
C) y(t) = e^(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s²-4) → inverse gives (1/2)sinh(2t).
Q286. Solve: y'(t)+2y(t)=δ(t), y(0)=0.
A) y(t) = e^(-2t) ✅
B) y(t) = e^(2t)
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s+2) → inverse gives e^(-2t).
Q287. Solve: y'(t)-2y(t)=δ(t), y(0)=0.
A) y(t) = e^(2t) ✅
B) y(t) = e^(-2t)
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s-2) → inverse gives e^(2t).
Q288. Solve: y''(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = 1 - cos(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Step input → transfer function 1/(s²+1) → output 1-cos(t).
Q289. Solve: y''(t)+2y'(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = t e^(-t) ✅
B) y(t) = e^(-t)
C) y(t) = 1 - e^(-t)
D) y(t) = sin(t)
Explanation: Step input → transfer function 1/(s+1)² → inverse gives t e^(-t).
Q290. Solve: y''(t)+4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1 - cos(2t)) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^(2t)
Explanation: Step input → transfer function 1/(s²+4) → inverse gives (1/2)(1-cos(2t)).
Q291. Solve: y''(t)-4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(cosh(2t) - 1) ✅
B) y(t) = sinh(2t)
C) y(t) = cosh(2t)
D) y(t) = e^(2t)
Explanation: Step input → transfer function 1/(s²-4) → inverse gives (1/4)(cosh(2t)-1).
Q292. Solve: y'(t)+y(t)=u(t), y(0)=0.
A) y(t) = 1 - e^(-t) ✅
B) y(t) = e^(-t)
C) y(t) = t
D) y(t) = sin(t)
Explanation: Step input → transfer function 1/(s+1) → inverse gives 1-e^(-t).
Q293. Solve: y'(t)-y(t)=u(t), y(0)=0.
A) y(t) = e^t - 1 ✅
B) y(t) = e^(-t)
C) y(t) = t
D) y(t) = sin(t)
Explanation: Step input → transfer function 1/(s-1) → inverse gives e^t-1.
Q294. Solve: y''(t)+y(t)=δ(t-a), y(0)=0, y'(0)=0.
A) y(t) = sin(t-a)u(t-a) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Shifted impulse → solution sin(t-a)u(t-a).
Q295. Solve: y''(t)+y(t)=u(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1 - cos(t-a))u(t-a) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Shifted step → solution (1 - cos(t-a))u(t-a).
Q296. Solve: y''(t)+4y(t)=δ(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sin(2(t-a)) u(t-a) ✅
B) y(t) = cos(2t)
C) y(t) = e^{2t}
D) y(t) = sinh(2t)
Explanation: Shifted impulse; transfer 1/(s²+4); inverse → (1/2)sin(2(t-a))u(t-a).
Q297. Solve: y''(t)+4y(t)=u(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1 - cos(2(t-a))) u(t-a) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^{2t}
Explanation: Shifted step; transfer 1/(s²+4); inverse → (1/2)(1 - cos(2(t-a)))u(t-a).
Q298. Solve: y''(t)-4y(t)=δ(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sinh(2(t-a)) u(t-a) ✅
B) y(t) = cosh(2t)
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted impulse; transfer 1/(s²-4); inverse → (1/2)sinh(2(t-a))u(t-a).
Q299. Solve: y''(t)-4y(t)=u(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(\cosh(2(t-a)) - 1) u(t-a) ✅
B) y(t) = \sinh(2t)
C) y(t) = \cosh(2t)
D) y(t) = e^{2t}
Explanation: Shifted step; transfer 1/(s²-4); inverse → (1/4)(cosh(2(t-a)) - 1)u(t-a).
Q300. Solve: y'(t)+y(t)=δ(t-a), y(0)=0.
A) y(t) = e^{-(t-a)} u(t-a) ✅
B) y(t) = e^{-t}
C) y(t) = e^{t}
D) y(t) = \sin t
Explanation: Shifted impulse; transfer 1/(s+1); inverse → e^{-(t-a)}u(t-a).
Q301. Solve: y'(t)+y(t)=u(t-a), y(0)=0.
A) y(t) = (1 - e^{-(t-a)}) u(t-a) ✅
B) y(t) = e^{-t}
C) y(t) = e^{t}
D) y(t) = \sin t
Explanation: Shifted step; transfer 1/(s+1); inverse → (1 - e^{-(t-a)})u(t-a).
Q302. Solve: y'(t)+2y(t)=δ(t-a), y(0)=0.
A) y(t) = e^{-2(t-a)} u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = \sin 2t
Explanation: Shifted impulse; transfer 1/(s+2); inverse → e^{-2(t-a)}u(t-a).
Q303. Solve: y'(t)+2y(t)=u(t-a), y(0)=0.
A) y(t) = (1 - e^{-2(t-a)}) u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = \sin 2t
Explanation: Shifted step; transfer 1/(s+2); inverse → (1 - e^{-2(t-a)})u(t-a).
Q304. Solve: y'(t)-2y(t)=δ(t-a), y(0)=0.
A) y(t) = e^{2(t-a)} u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = \sin 2t
Explanation: Shifted impulse; transfer 1/(s-2); inverse → e^{2(t-a)}u(t-a).
Q305. Solve: y'(t)-2y(t)=u(t-a), y(0)=0.
A) y(t) = (e^{2(t-a)} - 1) u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = \sin 2t
Explanation: Shifted step; transfer 1/(s-2); inverse → (e^{2(t-a)} - 1)u(t-a).
Q306. Solve: y''(t)+y(t)=t e^{-t}, y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{2}(t \sin t + (1 - \cos t)) e^{-t} ✅
B) y(t) = t e^{-t}
C) y(t) = \sin t
D) y(t) = \cos t
Explanation: Convolution with e^{-t}; partial fractions in s-domain → stated form.
Q307. Solve: y'(t)+y(t)=t e^{-t}, y(0)=0.
A) y(t) = \tfrac{1}{2} t^{2} e^{-t} ✅
B) y(t) = t e^{-t}
C) y(t) = e^{-t}
D) y(t) = 1 - e^{-t}
Explanation: Y(s)=1/(s+1)·1/(s+1)^{2} → 1/(s+1)^{3} → (t^{2}/2)e^{-t}.
Q308. Solve: y''(t)+2y'(t)+y(t)=t e^{-t}, y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{6} t^{3} e^{-t} ✅
B) y(t) = \tfrac{1}{2} t^{2} e^{-t}
C) y(t) = t e^{-t}
D) y(t) = e^{-t}
Explanation: Repeated pole at s=-1; Y(s)=1/(s+1)^{4} → (t^{3}/6)e^{-t}.
Q309. Solve: y''(t)+y(t)=e^{-t}\sin t, y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{2} e^{-t} ( \sin t - \cos t + \sin t \cos t ) ✅
B) y(t) = e^{-t}\sin t
C) y(t) = e^{-t}\cos t
D) y(t) = \sin t
Explanation: Product in time → shift/convolution in s; partial fractions → mixed term.
Q310. Solve: y'(t)+y(t)=e^{-t}\cos t, y(0)=0.
A) y(t) = \tfrac{1}{2} e^{-t} ( \cos t + \sin t ) ✅
B) y(t) = e^{-t}\cos t
C) y(t) = e^{-t}\sin t
D) y(t) = \cos t
Explanation: Y(s)= (s)/( (s+1)((s+1)^{2}+1) ); inverse → stated combination.
Q311. Solve: y''(t)+y(t)=u(t)\ast u(t) (convolution), y(0)=0, y'(0)=0.
A) y(t) = t - \sin t ✅
B) y(t) = 1 - \cos t
C) y(t) = \sin t
D) y(t) = \cos t
Explanation: u∗u = t·u(t); input t → output t - sin t.
Q312. Solve: y''(t)+4y(t)=u(t)\ast u(t), y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{2}( t - \sin 2t ) ✅
B) y(t) = \tfrac{1}{2}(1 - \cos 2t)
C) y(t) = \sin 2t
D) y(t) = \cos 2t
Explanation: u∗u = t; input t → output (1/2)(t - \sin 2t).
Q313. Solve: y''(t)-4y(t)=u(t)\ast u(t), y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{4}( \sinh 2t - 2t ) ✅
B) y(t) = \cosh 2t
C) y(t) = \sinh 2t
D) y(t) = e^{2t}
Explanation: Input t; transfer 1/(s²-4); inverse → (1/4)(\sinh 2t - 2t).
Q314. Solve: y''(t)+y(t)=\delta(t)\ast u(t), y(0)=0, y'(0)=0.
A) y(t) = 1 - \cos t ✅
B) y(t) = \sin t
C) y(t) = \cos t
D) y(t) = e^{t}
Explanation: δ∗u = u; step input → output 1 - \cos t.
Q315. Solve: y''(t)+y(t)=u(t)\ast \sin t, y(0)=0, y'(0)=0.
A) y(t) = t - \sin t - \tfrac{1}{2}(1 - \cos t) ✅
B) y(t) = \sin t
C) y(t) = 1 - \cos t
D) y(t) = t
Explanation: Convolution with sin t → s-domain product (1/s)(1/(s²+1)); partial fractions → stated form.
Q316. Solve: y''(t)+y(t)=u(t-Ï€), y(0)=0, y'(0)=0.
A) y(t) = (1 - cos(t-Ï€)) u(t-Ï€) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Shifted step input; transfer 1/(s²+1); inverse → (1 - cos(t-Ï€))u(t-Ï€).
Q317. Solve: y''(t)+y(t)=δ(t-π), y(0)=0, y'(0)=0.
A) y(t) = sin(t-Ï€) u(t-Ï€) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Shifted impulse; transfer 1/(s²+1); inverse → sin(t-Ï€)u(t-Ï€).
Q318. Solve: y''(t)+4y(t)=u(t-Ï€), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1 - cos(2(t-Ï€))) u(t-Ï€) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^{2t}
Explanation: Shifted step; transfer 1/(s²+4); inverse → (1/2)(1 - cos(2(t-Ï€)))u(t-Ï€).
Q319. Solve: y''(t)+4y(t)=δ(t-π), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sin(2(t-Ï€)) u(t-Ï€) ✅
B) y(t) = cos(2t)
C) y(t) = e^{2t}
D) y(t) = sinh(2t)
Explanation: Shifted impulse; transfer 1/(s²+4); inverse → (1/2)sin(2(t-Ï€))u(t-Ï€).
Q320. Solve: y''(t)-4y(t)=u(t-Ï€), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(cosh(2(t-Ï€)) - 1) u(t-Ï€) ✅
B) y(t) = sinh(2t)
C) y(t) = cosh(2t)
D) y(t) = e^{2t}
Explanation: Shifted step; transfer 1/(s²-4); inverse → (1/4)(cosh(2(t-Ï€))-1)u(t-Ï€).
Q321. Solve: y''(t)-4y(t)=δ(t-π), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sinh(2(t-Ï€)) u(t-Ï€) ✅
B) y(t) = cosh(2t)
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted impulse; transfer 1/(s²-4); inverse → (1/2)sinh(2(t-Ï€))u(t-Ï€).
Q322. Solve: y'(t)+y(t)=cos(t), y(0)=0.
A) y(t) = (1/2)(sin(t) - cos(t) + e^{-t}) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^{-t}
Explanation: Y(s)=s/[(s²+1)(s+1)]; partial fractions → stated form.
Q323. Solve: y'(t)+y(t)=sin(t), y(0)=0.
A) y(t) = (1/2)(1 - cos(t) - e^{-t}) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^{-t}
Explanation: Y(s)=1/[(s²+1)(s+1)]; partial fractions → stated form.
Q324. Solve: y''(t)+y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = t - sin(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Y(s)=1/[s²(s²+1)]; inverse → t - sin(t).
Q325. Solve: y''(t)-y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Y(s)=1/[s²(s²-1)]; inverse → sinh(t) - t.
Q326. Solve: y''(t)+2y'(t)+y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = t e^{-t} ✅
B) y(t) = e^{-t}
C) y(t) = 1 - e^{-t}
D) y(t) = sin(t)
Explanation: Transfer function 1/(s+1)²; inverse → t e^{-t}.
Q327. Solve: y''(t)+4y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = (1/2)(t - sin(2t)) ✅
B) y(t) = cos(2t)
C) y(t) = sin(2t)
D) y(t) = e^{2t}
Explanation: Y(s)=1/[s²(s²+4)]; inverse → (1/2)(t - sin(2t)).
Q328. Solve: y''(t)-4y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = (1/4)(sinh(2t) - 2t) ✅
B) y(t) = cosh(2t)
C) y(t) = sinh(2t)
D) y(t) = e^{2t}
Explanation: Y(s)=1/[s²(s²-4)]; inverse → (1/4)(sinh(2t) - 2t).
Q329. Solve: y'(t)+2y(t)=t, y(0)=0.
A) y(t) = t - 1/2 + (1/2)e^{-2t} ✅
B) y(t) = t
C) y(t) = e^{-2t}
D) y(t) = sin(t)
Explanation: Y(s)=1/[s²(s+2)]; partial fractions → stated form.
Q330. Solve: y'(t)+2y(t)=t², y(0)=0.
A) y(t) = t² - t + 1/2 - (1/2)e^{-2t} ✅
B) y(t) = t²
C) y(t) = e^{-2t}
D) y(t) = sin(t)
Explanation: Y(s)=2/[s³(s+2)]; partial fractions → stated form.
Q331. Solve: y''(t)+y(t)=t², y(0)=0, y'(0)=0.
A) y(t) = t² - 2 + 2cos(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Y(s)=2/[s³(s²+1)]; inverse → t² - 2 + 2cos(t).
Q332. Solve: y''(t)-y(t)=t², y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t² ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Y(s)=2/[s³(s²-1)]; inverse → sinh(t) - t².
Q333. Solve: y''(t)+y(t)=t³, y(0)=0, y'(0)=0.
A) y(t) = t³ - 6t + 6sin(t) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Y(s)=6/[s⁴(s²+1)]; inverse → t³ - 6t + 6sin(t).
Q334. Solve: y''(t)-y(t)=t³, y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t³ ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Y(s)=6/[s⁴(s²-1)] → inverse gives sinh(t)-t³.
Q335. Solve: y'(t)+2y(t)=t³, y(0)=0.
A) y(t) = t³ - 3t² + 6t - 6 + 6e^(-2t) ✅
B) y(t) = t³
C) y(t) = e^(-2t)
D) y(t) = sin(t)
Explanation: Y(s)=6/[s⁴(s+2)] → inverse gives polynomial minus exponential.
Q336. Solve: y''(t)+y(t)=t⁴, y(0)=0, y'(0)=0.
A) y(t) = t⁴ - 12t² + 24 - 24cos(t) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Y(s)=24/[s⁵(s²+1)] → inverse gives polynomial with cosine term.
Q337. Solve: y''(t)-y(t)=t⁴, y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t⁴ ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Y(s)=24/[s⁵(s²-1)] → inverse gives sinh(t)-t⁴.
Q338. Solve: y'(t)+y(t)=t⁴, y(0)=0.
A) y(t) = t⁴ - 4t³ + 12t² - 24t + 24 - 24e^(-t) ✅
B) y(t) = t⁴
C) y(t) = e^(-t)
D) y(t) = sin(t)
Explanation: Y(s)=24/[s⁵(s+1)] → inverse gives polynomial minus exponential.
Q339. Solve: y''(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = sin(t) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Impulse input → transfer function 1/(s²+1) → output sin(t).
Q340. Solve: y''(t)+2y'(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = e^(-t) ✅
B) y(t) = e^(t)
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Impulse response of system with (s+1)² denominator → e^(-t).
Q341. Solve: y''(t)+4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)sin(2t) ✅
B) y(t) = cos(2t)
C) y(t) = e^(2t)
D) y(t) = sinh(2t)
Explanation: Transfer function 1/(s²+4) → inverse gives (1/2)sin(2t).
Q342. Solve: y''(t)-4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)sinh(2t) ✅
B) y(t) = cosh(2t)
C) y(t) = e^(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s²-4) → inverse gives (1/2)sinh(2t).
Q343. Solve: y'(t)+2y(t)=δ(t), y(0)=0.
A) y(t) = e^(-2t) ✅
B) y(t) = e^(2t)
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s+2) → inverse gives e^(-2t).
Q344. Solve: y'(t)-2y(t)=δ(t), y(0)=0.
A) y(t) = e^(2t) ✅
B) y(t) = e^(-2t)
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s-2) → inverse gives e^(2t).
Q345. Solve: y''(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = 1 - cos(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Step input → transfer function 1/(s²+1) → output 1-cos(t).
Q346. Solve: y''(t)+2y'(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = t e^(-t) ✅
B) y(t) = e^(-t)
C) y(t) = 1 - e^(-t)
D) y(t) = sin(t)
Explanation: Step input → transfer function 1/(s+1)² → inverse gives t e^(-t).
Q347. Solve: y''(t)+4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1 - cos(2t)) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^(2t)
Explanation: Step input → transfer function 1/(s²+4) → inverse gives (1/2)(1-cos(2t)).
Q348. Solve: y''(t)-4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(cosh(2t) - 1) ✅
B) y(t) = sinh(2t)
C) y(t) = cosh(2t)
D) y(t) = e^(2t)
Explanation: Step input → transfer function 1/(s²-4) → inverse gives (1/4)(cosh(2t)-1).
Q349. Solve: y'(t)+y(t)=u(t), y(0)=0.
A) y(t) = 1 - e^(-t) ✅
B) y(t) = e^(-t)
C) y(t) = t
D) y(t) = sin(t)
Explanation: Step input → transfer function 1/(s+1) → inverse gives 1-e^(-t).
Q350. Solve: y'(t)-y(t)=u(t), y(0)=0.
A) y(t) = e^t - 1 ✅
B) y(t) = e^(-t)
C) y(t) = t
D) y(t) = sin(t)
Explanation: Step input → transfer function 1/(s-1) → inverse gives e^t-1.
Q351. Solve: y''(t)+y(t)=δ(t-a), y(0)=0, y'(0)=0.
A) y(t) = sin(t-a)u(t-a) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Shifted impulse → solution sin(t-a)u(t-a).
Q352. Solve: y''(t)+y(t)=u(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1 - cos(t-a))u(t-a) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Shifted step → solution (1 - cos(t-a))u(t-a).
Q353. Solve: y''(t)+4y(t)=δ(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sin(2(t-a)) u(t-a) ✅
B) y(t) = cos(2t)
C) y(t) = e^{2t}
D) y(t) = sinh(2t)
Explanation: Shifted impulse; transfer 1/(s²+4); inverse → (1/2)sin(2(t-a))u(t-a).
Q354. Solve: y''(t)+4y(t)=u(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1 - cos(2(t-a))) u(t-a) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^{2t}
Explanation: Shifted step; transfer 1/(s²+4); inverse → (1/2)(1 - cos(2(t-a)))u(t-a).
Q355. Solve: y''(t)-4y(t)=δ(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sinh(2(t-a)) u(t-a) ✅
B) y(t) = cosh(2t)
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted impulse; transfer 1/(s²-4); inverse → (1/2)sinh(2(t-a))u(t-a).
Q356. Solve: y''(t)-4y(t)=u(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(cosh(2(t-a)) - 1) u(t-a) ✅
B) y(t) = sinh(2t)
C) y(t) = cosh(2t)
D) y(t) = e^{2t}
Explanation: Shifted step; transfer 1/(s²-4); inverse → (1/4)(cosh(2(t-a))-1)u(t-a).
Q357. Solve: y'(t)+y(t)=δ(t-a), y(0)=0.
A) y(t) = e^{-(t-a)} u(t-a) ✅
B) y(t) = e^{-t}
C) y(t) = e^{t}
D) y(t) = sin(t)
Explanation: Shifted impulse; transfer 1/(s+1); inverse → e^{-(t-a)}u(t-a).
Q358. Solve: y'(t)+y(t)=u(t-a), y(0)=0.
A) y(t) = (1 - e^{-(t-a)}) u(t-a) ✅
B) y(t) = e^{-t}
C) y(t) = e^{t}
D) y(t) = sin(t)
Explanation: Shifted step; transfer 1/(s+1); inverse → (1 - e^{-(t-a)})u(t-a).
Q359. Solve: y'(t)+2y(t)=δ(t-a), y(0)=0.
A) y(t) = e^{-2(t-a)} u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted impulse; transfer 1/(s+2); inverse → e^{-2(t-a)}u(t-a).
Q360. Solve: y'(t)+2y(t)=u(t-a), y(0)=0.
A) y(t) = (1 - e^{-2(t-a)}) u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted step; transfer 1/(s+2); inverse → (1 - e^{-2(t-a)})u(t-a).
Q361. Solve: y'(t)-2y(t)=δ(t-a), y(0)=0.
A) y(t) = e^{2(t-a)} u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted impulse; transfer 1/(s-2); inverse → e^{2(t-a)}u(t-a).
Q362. Solve: y'(t)-2y(t)=u(t-a), y(0)=0.
A) y(t) = (e^{2(t-a)} - 1) u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted step; transfer 1/(s-2); inverse → (e^{2(t-a)} - 1)u(t-a).
Q363. Solve: y''(t)+y(t)=cos(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1 - cos(t) - t sin(t)) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Y(s)=s/[(s²+1)(s²+1)]; partial fractions → stated form.
Q364. Solve: y''(t)+y(t)=sin(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(sin(t) - t cos(t)) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Y(s)=1/[(s²+1)(s²+1)]; partial fractions → stated form.
Q365. Solve: y''(t)+4y(t)=cos(2t), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(1 - cos(2t) - t sin(2t)) ✅
B) y(t) = cos(2t)
C) y(t) = sin(2t)
D) y(t) = e^{2t}
Explanation: Resonant forcing; partial fractions → stated form.
Q366. Solve: y''(t)+4y(t)=sin(2t), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(sin(2t) - 2t cos(2t)) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^{2t}
Explanation: Resonant forcing; partial fractions → stated form.
Q367. Solve: y'(t)+2y(t)=cos(t), y(0)=0.
A) y(t) = (1/5)(2cos(t) + sin(t) - 2e^{-2t}) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^{-2t}
Explanation: Y(s)=s/[(s²+1)(s+2)]; partial fractions → stated form.
Q368. Solve: y'(t)+2y(t)=sin(t), y(0)=0.
A) y(t) = (1/5)(2sin(t) - cos(t) - e^{-2t}) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^{-2t}
Explanation: Y(s)=1/[(s²+1)(s+2)]; partial fractions → stated form.
Q369. Solve: y''(t)-y(t)=cosh(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(cosh(t) - sinh(t)) ✅
B) y(t) = cosh(t)
C) y(t) = sinh(t)
D) y(t) = e^t
Explanation: Y(s)=s/[(s²-1)(s²-1)]; partial fractions → stated form.
Q370. Solve: y''(t)-y(t)=sinh(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(sinh(t) - t) ✅
B) y(t) = sinh(t)
C) y(t) = cosh(t)
D) y(t) = e^t
Explanation: Y(s)=1/[(s²-1)(s²-1)]; inverse → stated form.
Q371. Solve: y''(t)+y(t)=e^{-t}, y(0)=0, y'(0)=0.
A) y(t) = (1/2)(e^{-t} - cos(t) + sin(t)) ✅
B) y(t) = e^{-t}
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Y(s)=1/[(s+1)(s²+1)]; partial fractions → stated form.
Q372. Solve: y'(t)+y(t)=e^{-t}, y(0)=0.
A) y(t) = t e^{-t} ✅
B) y(t) = e^{-t}
C) y(t) = 1 - e^{-t}
D) y(t) = sin(t)
Explanation: Y(s)=1/(s+1)²; inverse → t e^{-t}.
Q373. Solve: y''(t)+2y'(t)+y(t)=e^{-t}, y(0)=0, y'(0)=0.
A) y(t) = (1/2) t² e^{-t} ✅
B) y(t) = t e^{-t}
C) y(t) = e^{-t}
D) y(t) = sin(t)
Explanation: Repeated pole at s=-1; Y(s)=1/(s+1)³; inverse → (t²/2)e^{-t}.
Q374. Solve: y''(t)+y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = t - sin(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Y(s)=1/[s²(s²+1)]; inverse → t - sin(t).
Q375. Solve: y'(t)+y(t)=t, y(0)=0.
A) y(t) = t - 1 + e^{-t} ✅
B) y(t) = t
C) y(t) = e^{-t}
D) y(t) = sin(t)
Explanation: Y(s)=1/[s²(s+1)]; inverse → t - 1 + e^{-t}.
Q376. Solve: y''(t)-4y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = (1/4)(sinh(2t) - 2t) ✅
B) y(t) = cosh(2t)
C) y(t) = sinh(2t)
D) y(t) = e^{2t}
Explanation: Y(s)=1/[s²(s²-4)]; inverse → (1/4)(sinh(2t) - 2t).
Q377. Solve: y''(t)+4y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = (1/2)(t - sin(2t)) ✅
B) y(t) = cos(2t)
C) y(t) = sin(2t)
D) y(t) = e^{2t}
Explanation: Y(s)=1/[s²(s²+4)]; inverse → (1/2)(t - sin(2t)).
Q378. Solve: y'(t)+2y(t)=t, y(0)=0.
A) y(t) = t - 1/2 + (1/2)e^{-2t} ✅
B) y(t) = t
C) y(t) = e^{-2t}
D) y(t) = sin(t)
Explanation: Y(s)=1/[s²(s+2)]; partial fractions → stated form.
Q379. Solve: y'(t)+2y(t)=t², y(0)=0.
A) y(t) = t² - t + 1/2 - (1/2)e^{-2t} ✅
B) y(t) = t²
C) y(t) = e^{-2t}
D) y(t) = sin(t)
Explanation: Y(s)=2/[s³(s+2)]; partial fractions → stated form.
Q380. Solve: y''(t)+y(t)=t², y(0)=0, y'(0)=0.
A) y(t) = t² - 2 + 2cos(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Y(s)=2/[s³(s²+1)]; inverse → t² - 2 + 2cos(t).
Q381. Solve: y''(t)-y(t)=t², y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t² ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Y(s)=2/[s³(s²-1)]; inverse → sinh(t) - t².
Q382. Solve: y''(t)+y(t)=t³, y(0)=0, y'(0)=0.
A) y(t) = t³ - 6t + 6sin(t) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Y(s)=6/[s⁴(s²+1)]; inverse → t³ - 6t + 6sin(t).
Q383. Solve: y''(t)-y(t)=t³, y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t³ ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Y(s)=6/[s⁴(s²-1)]; inverse → sinh(t) - t³.
Q384. Solve: y'(t)+2y(t)=t³, y(0)=0.
A) y(t) = t³ - 3t² + 6t - 6 + 6e^{-2t} ✅
B) y(t) = t³
C) y(t) = e^{-2t}
D) y(t) = sin(t)
Explanation: Y(s)=6/[s⁴(s+2)]; inverse → polynomial minus exponential.
Q385. Solve: y''(t)+y(t)=t⁴, y(0)=0, y'(0)=0.
A) y(t) = t⁴ - 12t² + 24 - 24cos(t) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Y(s)=24/[s⁵(s²+1)]; inverse → polynomial with cosine term.
Q386. Solve: y''(t)-y(t)=t⁴, y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t⁴ ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Y(s)=24/[s⁵(s²-1)]; inverse → sinh(t) - t⁴.
Q387. Solve: y'(t)+y(t)=t⁴, y(0)=0.
A) y(t) = t⁴ - 4t³ + 12t² - 24t + 24 - 24e^{-t} ✅
B) y(t) = t⁴
C) y(t) = e^{-t}
D) y(t) = sin(t)
Explanation: Y(s)=24/[s⁵(s+1)]; inverse → polynomial minus exponential.
Q388. Solve: y''(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = sin(t) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Impulse input → transfer function 1/(s²+1) → output sin(t).
Q389. Solve: y''(t)+2y'(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = e^{-t} ✅
B) y(t) = e^t
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Impulse response of system with (s+1)² denominator → e^{-t}.
Q390. Solve: y''(t)+4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sin(2t) ✅
B) y(t) = cos(2t)
C) y(t) = e^{2t}
D) y(t) = sinh(2t)
Explanation: Transfer function 1/(s²+4); inverse → (1/2)sin(2t).
Q391. Solve: y''(t)-4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sinh(2t) ✅
B) y(t) = cosh(2t)
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s²-4); inverse → (1/2)sinh(2t).
Q392. Solve: y'(t)+2y(t)=δ(t), y(0)=0.
A) y(t) = e^{-2t} ✅
B) y(t) = e^{2t}
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s+2); inverse → e^{-2t}.
Q393. Solve: y'(t)-2y(t)=δ(t), y(0)=0.
A) y(t) = e^{2t} ✅
B) y(t) = e^{-2t}
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s-2); inverse → e^{2t}.
Q394. Solve: y''(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = 1 - cos(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Step input; transfer function 1/(s²+1); inverse → 1 - cos(t).
Q395. Solve: y''(t)+2y'(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = t e^{-t} ✅
B) y(t) = e^{-t}
C) y(t) = 1 - e^{-t}
D) y(t) = sin(t)
Explanation: Step input; transfer function 1/(s+1)²; inverse → t e^{-t}.
Q396. Solve: y''(t)+4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1 - cos(2t)) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^{2t}
Explanation: Step input; transfer function 1/(s²+4); inverse → (1/2)(1 - cos(2t)).
Q397. Solve: y''(t)-4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(cosh(2t) - 1) ✅
B) y(t) = sinh(2t)
C) y(t) = cosh(2t)
D) y(t) = e^{2t}
Explanation: Step input; transfer function 1/(s²-4); inverse → (1/4)(cosh(2t)-1).
Q398. Solve: y'(t)+y(t)=u(t), y(0)=0.
A) y(t) = 1 - e^{-t} ✅
B) y(t) = e^{-t}
C) y(t) = t
D) y(t) = sin(t)
Explanation: Step input; transfer function 1/(s+1); inverse → 1 - e^{-t}.
Q399. Solve: y'(t)-y(t)=u(t), y(0)=0.
A) y(t) = e^t - 1 ✅
B) y(t) = e^{-t}
C) y(t) = t
D) y(t) = sin(t)
Explanation: Step input; transfer function 1/(s-1); inverse → e^t - 1.
Q400. Solve: y''(t)+y(t)=δ(t-a), y(0)=0, y'(0)=0.
A) y(t) = sin(t-a) u(t-a) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Shifted impulse; transfer function 1/(s²+1); inverse → sin(t-a)u(t-a).
Q401. Solve: y''(t)+y(t)=u(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1 - cos(t-a)) u(t-a) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Shifted step; transfer function 1/(s²+1); inverse → (1 - cos(t-a))u(t-a).
Q402. Solve: y''(t)+4y(t)=δ(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sin(2(t-a)) u(t-a) ✅
B) y(t) = cos(2t)
C) y(t) = e^{2t}
D) y(t) = sinh(2t)
Explanation: Shifted impulse; transfer function 1/(s²+4); inverse → (1/2)sin(2(t-a))u(t-a).
Q403. Solve: y''(t)+4y(t)=u(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1 - cos(2(t-a))) u(t-a) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^{2t}
Explanation: Shifted step; transfer function 1/(s²+4); inverse → (1/2)(1 - cos(2(t-a)))u(t-a).
Q404. Solve: y''(t)-4y(t)=δ(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sinh(2(t-a)) u(t-a) ✅
B) y(t) = cosh(2t)
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted impulse; transfer function 1/(s²-4); inverse → (1/2)sinh(2(t-a))u(t-a).
Q405. Solve: y''(t)-4y(t)=u(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(cosh(2(t-a)) - 1) u(t-a) ✅
B) y(t) = sinh(2t)
C) y(t) = cosh(2t)
D) y(t) = e^{2t}
Explanation: Shifted step; transfer function 1/(s²-4); inverse → (1/4)(cosh(2(t-a))-1)u(t-a).
Q406. Solve: y'(t)+2y(t)=δ(t-a), y(0)=0.
A) y(t) = e^{-2(t-a)} u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted impulse; transfer function 1/(s+2); inverse → e^{-2(t-a)}u(t-a).
Q407. Solve: y'(t)+2y(t)=u(t-a), y(0)=0.
A) y(t) = (1 - e^{-2(t-a)}) u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted step; transfer function 1/(s+2); inverse → (1 - e^{-2(t-a)})u(t-a).
Q408. Solve: y'(t)-2y(t)=δ(t-a), y(0)=0.
A) y(t) = e^{2(t-a)} u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted impulse; transfer 1/(s-2); inverse → e^{2(t-a)}u(t-a).
Q409. Solve: y'(t)-2y(t)=u(t-a), y(0)=0.
A) y(t) = (e^{2(t-a)} - 1) u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted step; transfer 1/(s-2); inverse → (e^{2(t-a)} - 1)u(t-a).
Q410. Solve: y''(t)+y(t)=cos(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1 - cos(t) - t sin(t)) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Y(s)=s/[(s²+1)(s²+1)]; partial fractions → stated form.
Q411. Solve: y''(t)+y(t)=sin(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(sin(t) - t cos(t)) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Y(s)=1/[(s²+1)(s²+1)]; partial fractions → stated form.
Q412. Solve: y''(t)+4y(t)=cos(2t), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(1 - cos(2t) - t sin(2t)) ✅
B) y(t) = cos(2t)
C) y(t) = sin(2t)
D) y(t) = e^{2t}
Explanation: Resonant forcing; partial fractions → stated form.
Q413. Solve: y''(t)+4y(t)=sin(2t), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(sin(2t) - 2t cos(2t)) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^{2t}
Explanation: Resonant forcing; partial fractions → stated form.
Q414. Solve: y'(t)+2y(t)=cos(t), y(0)=0.
A) y(t) = (1/5)(2cos(t) + sin(t) - 2e^{-2t}) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^{-2t}
Explanation: Y(s)=s/[(s²+1)(s+2)]; partial fractions → stated form.
Q415. Solve: y'(t)+2y(t)=sin(t), y(0)=0.
A) y(t) = (1/5)(2sin(t) - cos(t) - e^{-2t}) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^{-2t}
Explanation: Y(s)=1/[(s²+1)(s+2)]; partial fractions → stated form.
Q416. Solve: y''(t)-y(t)=cosh(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(cosh(t) - sinh(t)) ✅
B) y(t) = cosh(t)
C) y(t) = sinh(t)
D) y(t) = e^t
Explanation: Y(s)=s/[(s²-1)(s²-1)]; partial fractions → stated form.
Q417. Solve: y''(t)-y(t)=sinh(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(sinh(t) - t) ✅
B) y(t) = sinh(t)
C) y(t) = cosh(t)
D) y(t) = e^t
Explanation: Y(s)=1/[(s²-1)(s²-1)]; inverse → stated form.
Q418. Solve: y''(t)+y(t)=e^{-t}, y(0)=0, y'(0)=0.
A) y(t) = (1/2)(e^{-t} - cos(t) + sin(t)) ✅
B) y(t) = e^{-t}
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Y(s)=1/[(s+1)(s²+1)]; partial fractions → stated form.
Q419. Solve: y'(t)+y(t)=e^{-t}, y(0)=0.
A) y(t) = t e^{-t} ✅
B) y(t) = e^{-t}
C) y(t) = 1 - e^{-t}
D) y(t) = sin(t)
Explanation: Y(s)=1/(s+1)²; inverse → t e^{-t}.
Q420. Solve: y''(t)+2y'(t)+y(t)=e^{-t}, y(0)=0, y'(0)=0.
A) y(t) = (1/2) t² e^{-t} ✅
B) y(t) = t e^{-t}
C) y(t) = e^{-t}
D) y(t) = sin(t)
Explanation: Repeated pole at s=-1; Y(s)=1/(s+1)³; inverse → (t²/2)e^{-t}.
Q421. Solve: y''(t)+y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = t - sin(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Y(s)=1/[s²(s²+1)]; inverse → t - sin(t).
Q422. Solve: y'(t)+y(t)=t, y(0)=0.
A) y(t) = t - 1 + e^{-t} ✅
B) y(t) = t
C) y(t) = e^{-t}
D) y(t) = sin(t)
Explanation: Y(s)=1/[s²(s+1)]; inverse → t - 1 + e^{-t}.
Q423. Solve: y''(t)-4y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = (1/4)(sinh(2t) - 2t) ✅
B) y(t) = cosh(2t)
C) y(t) = sinh(2t)
D) y(t) = e^{2t}
Explanation: Y(s)=1/[s²(s²-4)]; inverse → (1/4)(sinh(2t) - 2t).
Q424. Solve: y''(t)+4y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = (1/2)(t - sin(2t)) ✅
B) y(t) = cos(2t)
C) y(t) = sin(2t)
D) y(t) = e^{2t}
Explanation: Y(s)=1/[s²(s²+4)]; inverse → (1/2)(t - sin(2t)).
Q425. Solve: y'(t)+2y(t)=t, y(0)=0.
A) y(t) = t - 1/2 + (1/2)e^{-2t} ✅
B) y(t) = t
C) y(t) = e^{-2t}
D) y(t) = sin(t)
Explanation: Y(s)=1/[s²(s+2)]; partial fractions → stated form.
Q426. Solve: y'(t)+2y(t)=t², y(0)=0.
A) y(t) = t² - t + 1/2 - (1/2)e^{-2t} ✅
B) y(t) = t²
C) y(t) = e^{-2t}
D) y(t) = sin(t)
Explanation: Y(s)=2/[s³(s+2)]; partial fractions → stated form.
Q427. Solve: y''(t)+y(t)=t², y(0)=0, y'(0)=0.
A) y(t) = t² - 2 + 2cos(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Y(s)=2/[s³(s²+1)]; inverse → t² - 2 + 2cos(t).
Q428. Solve: y''(t)-y(t)=t², y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t² ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Y(s)=2/[s³(s²-1)]; inverse → sinh(t) - t².
Q429. Solve: y''(t)+y(t)=t³, y(0)=0, y'(0)=0.
A) y(t) = t³ - 6t + 6sin(t) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Y(s)=6/[s⁴(s²+1)]; inverse → t³ - 6t + 6sin(t).
Q430. Solve: y''(t)-y(t)=t³, y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t³ ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Y(s)=6/[s⁴(s²-1)]; inverse → sinh(t) - t³.
Q431. Solve: y'(t)+2y(t)=t³, y(0)=0.
A) y(t) = t³ - 3t² + 6t - 6 + 6e^{-2t} ✅
B) y(t) = t³
C) y(t) = e^{-2t}
D) y(t) = sin(t)
Explanation: Y(s)=6/[s⁴(s+2)]; inverse → polynomial minus exponential.
Q432. Solve: y''(t)+y(t)=t⁴, y(0)=0, y'(0)=0.
A) y(t) = t⁴ - 12t² + 24 - 24cos(t) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Y(s)=24/[s⁵(s²+1)]; inverse → polynomial with cosine term.
Q433. Solve: y''(t)-y(t)=t⁴, y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t⁴ ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Y(s)=24/[s⁵(s²-1)]; inverse → sinh(t) - t⁴.
Q434. Solve: y'(t)+y(t)=t⁴, y(0)=0.
A) y(t) = t⁴ - 4t³ + 12t² - 24t + 24 - 24e^{-t} ✅
B) y(t) = t⁴
C) y(t) = e^{-t}
D) y(t) = sin(t)
Explanation: Y(s)=24/[s⁵(s+1)]; inverse → polynomial minus exponential.
Q435. Solve: y''(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = sin(t) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Impulse input; transfer function 1/(s²+1); output sin(t).
Q436. Solve: y''(t)+2y'(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = e^{-t} ✅
B) y(t) = e^t
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Impulse response of system with (s+1)² denominator → e^{-t}.
Q437. Solve: y''(t)+4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sin(2t) ✅
B) y(t) = cos(2t)
C) y(t) = e^{2t}
D) y(t) = sinh(2t)
Explanation: Transfer function 1/(s²+4); inverse → (1/2)sin(2t).
Q438. Solve: y''(t)-4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sinh(2t) ✅
B) y(t) = cosh(2t)
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s²-4); inverse → (1/2)sinh(2t).
Q439. Solve: y'(t)+2y(t)=δ(t), y(0)=0.
A) y(t) = e^{-2t} ✅
B) y(t) = e^{2t}
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s+2); inverse → e^{-2t}.
Q440. Solve: y'(t)-2y(t)=δ(t), y(0)=0.
A) y(t) = e^{2t} ✅
B) y(t) = e^{-2t}
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s-2); inverse → e^{2t}.
Q441. Solve: y''(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = 1 - cos(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Step input; transfer function 1/(s²+1); inverse → 1 - cos(t).
Q442. Solve: y''(t)+2y'(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = t e^{-t} ✅
B) y(t) = e^{-t}
C) y(t) = 1 - e^{-t}
D) y(t) = sin(t)
Explanation: Step input; transfer function 1/(s+1)²; inverse → t e^{-t}.
Q443. Solve: y''(t)+4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1 - cos(2t)) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^{2t}
Explanation: Step input; transfer function 1/(s²+4); inverse → (1/2)(1 - cos(2t)).
Q444. Solve: y''(t)-4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(cosh(2t) - 1) ✅
B) y(t) = sinh(2t)
C) y(t) = cosh(2t)
D) y(t) = e^{2t}
Explanation: Step input; transfer function 1/(s²-4); inverse → (1/4)(cosh(2t)-1).
Q445. Solve: y'(t)+y(t)=u(t), y(0)=0.
A) y(t) = 1 - e^{-t} ✅
B) y(t) = e^{-t}
C) y(t) = t
D) y(t) = sin(t)
Explanation: Step input; transfer function 1/(s+1); inverse → 1 - e^{-t}.
Q446. Solve: y'(t)-y(t)=u(t), y(0)=0.
A) y(t) = e^t - 1 ✅
B) y(t) = e^{-t}
C) y(t) = t
D) y(t) = sin(t)
Explanation: Step input; transfer function 1/(s-1); inverse → e^t - 1.
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Q447. Solve: y''(t)+y(t)=δ(t-a), y(0)=0, y'(0)=0.
A) y(t) = \sin(t-a)\,u(t-a) ✅
B) y(t) = \cos(t)
C) y(t) = e^{t}
D) y(t) = \sinh(t)
Explanation: Shifted impulse; transfer 1/(s²+1); inverse → \sin(t-a)u(t-a).
Q448. Solve: y''(t)+y(t)=u(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1 - \cos(t-a))\,u(t-a) ✅
B) y(t) = \sin(t)
C) y(t) = \cos(t)
D) y(t) = e^{t}
Explanation: Shifted step; transfer 1/(s²+1); inverse → (1 - \cos(t-a))u(t-a).
Q449. Solve: y''(t)+4y(t)=δ(t-a), y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{2}\sin(2(t-a))\,u(t-a) ✅
B) y(t) = \cos(2t)
C) y(t) = e^{2t}
D) y(t) = \sinh(2t)
Explanation: Shifted impulse; transfer 1/(s²+4); inverse → (1/2)\sin(2(t-a))u(t-a).
Q450. Solve: y''(t)+4y(t)=u(t-a), y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{2}(1 - \cos(2(t-a)))\,u(t-a) ✅
B) y(t) = \sin(2t)
C) y(t) = \cos(2t)
D) y(t) = e^{2t}
Explanation: Shifted step; transfer 1/(s²+4); inverse → (1/2)(1 - \cos(2(t-a)))u(t-a).
Q451. Solve: y''(t)-4y(t)=δ(t-a), y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{2}\sinh(2(t-a))\,u(t-a) ✅
B) y(t) = \cosh(2t)
C) y(t) = e^{2t}
D) y(t) = \sin(2t)
Explanation: Shifted impulse; transfer 1/(s²-4); inverse → (1/2)\sinh(2(t-a))u(t-a).
Q452. Solve: y''(t)-4y(t)=u(t-a), y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{4}(\cosh(2(t-a)) - 1)\,u(t-a) ✅
B) y(t) = \sinh(2t)
C) y(t) = \cosh(2t)
D) y(t) = e^{2t}
Explanation: Shifted step; transfer 1/(s²-4); inverse → (1/4)(\cosh(2(t-a)) - 1)u(t-a).
Q453. Solve: y'(t)+y(t)=δ(t-a), y(0)=0.
A) y(t) = e^{-(t-a)}\,u(t-a) ✅
B) y(t) = e^{-t}
C) y(t) = e^{t}
D) y(t) = \sin t
Explanation: Shifted impulse; transfer 1/(s+1); inverse → e^{-(t-a)}u(t-a).
Q454. Solve: y'(t)+y(t)=u(t-a), y(0)=0.
A) y(t) = (1 - e^{-(t-a)})\,u(t-a) ✅
B) y(t) = e^{-t}
C) y(t) = e^{t}
D) y(t) = \sin t
Explanation: Shifted step; transfer 1/(s+1); inverse → (1 - e^{-(t-a)})u(t-a).
Q455. Solve: y'(t)+2y(t)=δ(t-a), y(0)=0.
A) y(t) = e^{-2(t-a)}\,u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = \sin 2t
Explanation: Shifted impulse; transfer 1/(s+2); inverse → e^{-2(t-a)}u(t-a).
Q456. Solve: y'(t)+2y(t)=u(t-a), y(0)=0.
A) y(t) = (1 - e^{-2(t-a)})\,u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = \sin 2t
Explanation: Shifted step; transfer 1/(s+2); inverse → (1 - e^{-2(t-a)})u(t-a).
Q457. Solve: y'(t)-2y(t)=δ(t-a), y(0)=0.
A) y(t) = e^{2(t-a)}\,u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = \sin 2t
Explanation: Shifted impulse; transfer 1/(s-2); inverse → e^{2(t-a)}u(t-a).
Q458. Solve: y'(t)-2y(t)=u(t-a), y(0)=0.
A) y(t) = (e^{2(t-a)} - 1)\,u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = \sin 2t
Explanation: Shifted step; transfer 1/(s-2); inverse → (e^{2(t-a)} - 1)u(t-a).
Q459. Solve: y''(t)+y(t)=\cos t, y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{2}(1 - \cos t - t \sin t) ✅
B) y(t) = \cos t
C) y(t) = \sin t
D) y(t) = e^{t}
Explanation: Y(s)=s/[(s²+1)²]; partial fractions → stated form.
Q460. Solve: y''(t)+y(t)=\sin t, y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{2}(\sin t - t \cos t) ✅
B) y(t) = \sin t
C) y(t) = \cos t
D) y(t) = e^{t}
Explanation: Y(s)=1/[(s²+1)²]; partial fractions → stated form.
Q461. Solve: y''(t)+4y(t)=\cos 2t, y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{4}(1 - \cos 2t - t \sin 2t) ✅
B) y(t) = \cos 2t
C) y(t) = \sin 2t
D) y(t) = e^{2t}
Explanation: Resonant forcing; repeated factor (s²+4); yields polynomial–trig mix.
Q462. Solve: y''(t)+4y(t)=\sin 2t, y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{4}(\sin 2t - 2t \cos 2t) ✅
B) y(t) = \sin 2t
C) y(t) = \cos 2t
D) y(t) = e^{2t}
Explanation: Resonant forcing; partial fractions → stated form.
Q463. Solve: y'(t)+2y(t)=\cos t, y(0)=0.
A) y(t) = \tfrac{1}{5}(2\cos t + \sin t - 2e^{-2t}) ✅
B) y(t) = \cos t
C) y(t) = \sin t
D) y(t) = e^{-2t}
Explanation: Y(s)=s/[(s²+1)(s+2)]; partial fractions → stated form.
Q464. Solve: y'(t)+2y(t)=\sin t, y(0)=0.
A) y(t) = \tfrac{1}{5}(2\sin t - \cos t - e^{-2t}) ✅
B) y(t) = \sin t
C) y(t) = \cos t
D) y(t) = e^{-2t}
Explanation: Y(s)=1/[(s²+1)(s+2)]; partial fractions → stated form.
Q465. Solve: y''(t)-y(t)=\cosh t, y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{2}(\cosh t - \sinh t) ✅
B) y(t) = \cosh t
C) y(t) = \sinh t
D) y(t) = e^{t}
Explanation: Y(s)=s/[(s²-1)²]; partial fractions → stated form.
Q466. Solve: y''(t)-y(t)=\sinh t, y(0)=0, y'(0)=0.
A) y(t) = \tfrac{1}{2}(\sinh t - t) ✅
B) y(t) = \sinh t
C) y(t) = \cosh t
D) y(t) = e^{t}
Explanation: Y(s)=1/[(s²-1)²]; inverse → stated form.
Q467. Solve: y''(t)+y(t)=e^{-t}, y(0)=0, y'(0)=0.
A) y(t) = (1/2)(e^{-t} - cos(t) + sin(t)) ✅
B) y(t) = e^{-t}
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Y(s)=1/[(s+1)(s²+1)]; partial fractions → stated form.
Q468. Solve: y'(t)+y(t)=e^{-t}, y(0)=0.
A) y(t) = t e^{-t} ✅
B) y(t) = e^{-t}
C) y(t) = 1 - e^{-t}
D) y(t) = sin(t)
Explanation: Y(s)=1/(s+1)²; inverse → t e^{-t}.
Q469. Solve: y''(t)+2y'(t)+y(t)=e^{-t}, y(0)=0, y'(0)=0.
A) y(t) = (1/2) t² e^{-t} ✅
B) y(t) = t e^{-t}
C) y(t) = e^{-t}
D) y(t) = sin(t)
Explanation: Repeated pole at s=-1; Y(s)=1/(s+1)³; inverse → (t²/2)e^{-t}.
Q470. Solve: y''(t)+y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = t - sin(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Y(s)=1/[s²(s²+1)]; inverse → t - sin(t).
Q471. Solve: y'(t)+y(t)=t, y(0)=0.
A) y(t) = t - 1 + e^{-t} ✅
B) y(t) = t
C) y(t) = e^{-t}
D) y(t) = sin(t)
Explanation: Y(s)=1/[s²(s+1)]; inverse → t - 1 + e^{-t}.
Q472. Solve: y''(t)-4y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = (1/4)(sinh(2t) - 2t) ✅
B) y(t) = cosh(2t)
C) y(t) = sinh(2t)
D) y(t) = e^{2t}
Explanation: Y(s)=1/[s²(s²-4)]; inverse → (1/4)(sinh(2t) - 2t).
Q473. Solve: y''(t)+4y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = (1/2)(t - sin(2t)) ✅
B) y(t) = cos(2t)
C) y(t) = sin(2t)
D) y(t) = e^{2t}
Explanation: Y(s)=1/[s²(s²+4)]; inverse → (1/2)(t - sin(2t)).
Q474. Solve: y'(t)+2y(t)=t, y(0)=0.
A) y(t) = t - 1/2 + (1/2)e^{-2t} ✅
B) y(t) = t
C) y(t) = e^{-2t}
D) y(t) = sin(t)
Explanation: Y(s)=1/[s²(s+2)]; partial fractions → stated form.
Q475. Solve: y'(t)+2y(t)=t², y(0)=0.
A) y(t) = t² - t + 1/2 - (1/2)e^{-2t} ✅
B) y(t) = t²
C) y(t) = e^{-2t}
D) y(t) = sin(t)
Explanation: Y(s)=2/[s³(s+2)]; partial fractions → stated form.
Q476. Solve: y''(t)+y(t)=t², y(0)=0, y'(0)=0.
A) y(t) = t² - 2 + 2cos(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Y(s)=2/[s³(s²+1)]; inverse → t² - 2 + 2cos(t).
Q477. Solve: y''(t)-y(t)=t², y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t² ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Y(s)=2/[s³(s²-1)]; inverse → sinh(t) - t².
Q478. Solve: y''(t)+y(t)=t³, y(0)=0, y'(0)=0.
A) y(t) = t³ - 6t + 6sin(t) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Y(s)=6/[s⁴(s²+1)]; inverse → t³ - 6t + 6sin(t).
Q479. Solve: y''(t)-y(t)=t³, y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t³ ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Y(s)=6/[s⁴(s²-1)]; inverse → sinh(t) - t³.
Q480. Solve: y'(t)+2y(t)=t³, y(0)=0.
A) y(t) = t³ - 3t² + 6t - 6 + 6e^{-2t} ✅
B) y(t) = t³
C) y(t) = e^{-2t}
D) y(t) = sin(t)
Explanation: Y(s)=6/[s⁴(s+2)]; inverse → polynomial minus exponential.
Q481. Solve: y''(t)+y(t)=t⁴, y(0)=0, y'(0)=0.
A) y(t) = t⁴ - 12t² + 24 - 24cos(t) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Y(s)=24/[s⁵(s²+1)]; inverse → polynomial with cosine term.
Q482. Solve: y''(t)-y(t)=t⁴, y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t⁴ ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Y(s)=24/[s⁵(s²-1)]; inverse → sinh(t) - t⁴.
Q483. Solve: y'(t)+y(t)=t⁴, y(0)=0.
A) y(t) = t⁴ - 4t³ + 12t² - 24t + 24 - 24e^{-t} ✅
B) y(t) = t⁴
C) y(t) = e^{-t}
D) y(t) = sin(t)
Explanation: Y(s)=24/[s⁵(s+1)]; inverse → polynomial minus exponential.
Q484. Solve: y''(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = sin(t) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Impulse input; transfer function 1/(s²+1); output sin(t).
Q485. Solve: y''(t)+2y'(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = e^{-t} ✅
B) y(t) = e^t
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Impulse response of system with (s+1)² denominator → e^{-t}.
Q486. Solve: y''(t)+4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sin(2t) ✅
B) y(t) = cos(2t)
C) y(t) = e^{2t}
D) y(t) = sinh(2t)
Explanation: Transfer function 1/(s²+4); inverse → (1/2)sin(2t).
Q487. Solve: y''(t)-4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sinh(2t) ✅
B) y(t) = cosh(2t)
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s²-4); inverse → (1/2)sinh(2t).
Q488. Solve: y'(t)+2y(t)=δ(t), y(0)=0.
A) y(t) = e^{-2t} ✅
B) y(t) = e^{2t}
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s+2); inverse → e^{-2t}.
Q489. Solve: y'(t)-2y(t)=δ(t), y(0)=0.
A) y(t) = e^{2t} ✅
B) y(t) = e^{-2t}
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s-2); inverse → e^{2t}.
Q490. Solve: y''(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = 1 - cos(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Step input; transfer function 1/(s²+1); inverse → 1 - cos(t).
Q491. Solve: y''(t)+2y'(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = t e^{-t} ✅
B) y(t) = e^{-t}
C) y(t) = 1 - e^{-t}
D) y(t) = sin(t)
Explanation: Step input; transfer function 1/(s+1)²; inverse → t e^{-t}.
Q492. Solve: y''(t)+4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1 - cos(2t)) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^{2t}
Explanation: Step input; transfer function 1/(s²+4); inverse → (1/2)(1 - cos(2t)).
Q493. Solve: y''(t)-4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(cosh(2t) - 1) ✅
B) y(t) = sinh(2t)
C) y(t) = cosh(2t)
D) y(t) = e^{2t}
Explanation: Step input; transfer function 1/(s²-4); inverse → (1/4)(cosh(2t)-1).
Q494. Solve: y'(t)+y(t)=u(t), y(0)=0.
A) y(t) = 1 - e^{-t} ✅
B) y(t) = e^{-t}
C) y(t) = t
D) y(t) = sin(t)
Explanation: Step input; transfer function 1/(s+1); inverse → 1 - e^{-t}.
Q495. Solve: y'(t)-y(t)=u(t), y(0)=0.
A) y(t) = e^t - 1 ✅
B) y(t) = e^{-t}
C) y(t) = t
D) y(t) = sin(t)
Explanation: Step input; transfer function 1/(s-1); inverse → e^t - 1.
Q496. Solve: y''(t)+y(t)=δ(t-a), y(0)=0, y'(0)=0.
A) y(t) = sin(t-a) u(t-a) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Shifted impulse; transfer function 1/(s²+1); inverse → sin(t-a)u(t-a).
Q497. Solve: y''(t)+y(t)=u(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1 - cos(t-a)) u(t-a) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Shifted step; transfer function 1/(s²+1); inverse → (1 - cos(t-a))u(t-a).
Q498. Solve: y''(t)+4y(t)=δ(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sin(2(t-a)) u(t-a) ✅
B) y(t) = cos(2t)
C) y(t) = e^{2t}
D) y(t) = sinh(2t)
Explanation: Shifted impulse; transfer function 1/(s²+4); inverse → (1/2)sin(2(t-a))u(t-a).
Q499. Solve: y''(t)+4y(t)=u(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1 - cos(2(t-a))) u(t-a) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^{2t}
Explanation: Shifted step; transfer function 1/(s²+4); inverse → (1/2)(1 - cos(2(t-a)))u(t-a).
Q500. Solve: y''(t)-4y(t)=δ(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sinh(2(t-a)) u(t-a) ✅
B) y(t) = cosh(2t)
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted impulse; transfer function 1/(s²-4); inverse → (1/2)sinh(2(t-a))u(t-a).
Q501. Solve: y''(t)-4y(t)=u(t-a), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(cosh(2(t-a)) - 1) u(t-a) ✅
B) y(t) = sinh(2t)
C) y(t) = cosh(2t)
D) y(t) = e^{2t}
Explanation: Shifted step; transfer function 1/(s²-4); inverse → (1/4)(cosh(2(t-a))-1)u(t-a).
Q502. Solve: y'(t)+2y(t)=δ(t-a), y(0)=0.
A) y(t) = e^{-2(t-a)} u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted impulse; transfer function 1/(s+2); inverse → e^{-2(t-a)}u(t-a).
Q503. Solve: y'(t)+2y(t)=u(t-a), y(0)=0.
A) y(t) = (1 - e^{-2(t-a)}) u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted step; transfer function 1/(s+2); inverse → (1 - e^{-2(t-a)})u(t-a).
Q504. Solve: y'(t)-2y(t)=δ(t-a), y(0)=0.
A) y(t) = e^{2(t-a)} u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted impulse; transfer 1/(s-2); inverse → e^{2(t-a)}u(t-a).
Q505. Solve: y'(t)-2y(t)=u(t-a), y(0)=0.
A) y(t) = (e^{2(t-a)} - 1) u(t-a) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted step; transfer 1/(s-2); inverse → (e^{2(t-a)} - 1)u(t-a).
Q506. Solve: y''(t)+y(t)=cos(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1 - cos(t) - t sin(t)) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Y(s)=s/[(s²+1)²]; partial fractions → stated form.
Q507. Solve: y''(t)+y(t)=sin(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(sin(t) - t cos(t)) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Y(s)=1/[(s²+1)²]; partial fractions → stated form.
Q508. Solve: y''(t)+4y(t)=cos(2t), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(1 - cos(2t) - t sin(2t)) ✅
B) y(t) = cos(2t)
C) y(t) = sin(2t)
D) y(t) = e^{2t}
Explanation: Resonant forcing; repeated factor (s²+4); yields polynomial–trig mix.
Q509. Solve: y''(t)+4y(t)=sin(2t), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(sin(2t) - 2t cos(2t)) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^{2t}
Explanation: Resonant forcing; partial fractions → stated form.
Q510. Solve: y'(t)+2y(t)=cos(t), y(0)=0.
A) y(t) = (1/5)(2cos(t) + sin(t) - 2e^{-2t}) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^{-2t}
Explanation: Y(s)=s/[(s²+1)(s+2)]; partial fractions → stated form.
Q511. Solve: y'(t)+2y(t)=sin(t), y(0)=0.
A) y(t) = (1/5)(2sin(t) - cos(t) - e^{-2t}) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^{-2t}
Explanation: Y(s)=1/[(s²+1)(s+2)]; partial fractions → stated form.
Q512. Solve: y''(t)-y(t)=cosh(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(cosh(t) - sinh(t)) ✅
B) y(t) = cosh(t)
C) y(t) = sinh(t)
D) y(t) = e^t
Explanation: Y(s)=s/[(s²-1)²]; partial fractions → stated form.
Q513. Solve: y''(t)-y(t)=sinh(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(sinh(t) - t) ✅
B) y(t) = sinh(t)
C) y(t) = cosh(t)
D) y(t) = e^t
Explanation: Y(s)=1/[(s²-1)²]; inverse → stated form.
Q514. Solve: y''(t)+y(t)=e^{-t}, y(0)=0, y'(0)=0.
A) y(t) = (1/2)(e^{-t} - cos(t) + sin(t)) ✅
B) y(t) = e^{-t}
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Y(s)=1/[(s+1)(s²+1)]; partial fractions → stated form.
Q515. Solve: y'(t)+y(t)=e^{-t}, y(0)=0.
A) y(t) = t e^{-t} ✅
B) y(t) = e^{-t}
C) y(t) = 1 - e^{-t}
D) y(t) = sin(t)
Explanation: Y(s)=1/(s+1)²; inverse → t e^{-t}.
Q516. Solve: y''(t)+2y'(t)+y(t)=e^{-t}, y(0)=0, y'(0)=0.
A) y(t) = (1/2) t² e^{-t} ✅
B) y(t) = t e^{-t}
C) y(t) = e^{-t}
D) y(t) = sin(t)
Explanation: Repeated pole at s=-1; Y(s)=1/(s+1)³; inverse → (t²/2)e^{-t}.
Q517. Solve: y''(t)+y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = t - sin(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Y(s)=1/[s²(s²+1)]; inverse → t - sin(t).
Q518. Solve: y'(t)+y(t)=t, y(0)=0.
A) y(t) = t - 1 + e^{-t} ✅
B) y(t) = t
C) y(t) = e^{-t}
D) y(t) = sin(t)
Explanation: Y(s)=1/[s²(s+1)]; inverse → t - 1 + e^{-t}.
Q519. Solve: y''(t)-4y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = (1/4)(sinh(2t) - 2t) ✅
B) y(t) = cosh(2t)
C) y(t) = sinh(2t)
D) y(t) = e^{2t}
Explanation: Y(s)=1/[s²(s²-4)]; inverse → (1/4)(sinh(2t) - 2t).
Q520. Solve: y''(t)+4y(t)=t, y(0)=0, y'(0)=0.
A) y(t) = (1/2)(t - sin(2t)) ✅
B) y(t) = cos(2t)
C) y(t) = sin(2t)
D) y(t) = e^{2t}
Explanation: Y(s)=1/[s²(s²+4)]; inverse → (1/2)(t - sin(2t)).
Q521. Solve: y'(t)+2y(t)=t, y(0)=0.
A) y(t) = t - 1/2 + (1/2)e^{-2t} ✅
B) y(t) = t
C) y(t) = e^{-2t}
D) y(t) = sin(t)
Explanation: Y(s)=1/[s²(s+2)]; partial fractions → stated form.
Q522. Solve: y'(t)+2y(t)=t², y(0)=0.
A) y(t) = t² - t + 1/2 - (1/2)e^{-2t} ✅
B) y(t) = t²
C) y(t) = e^{-2t}
D) y(t) = sin(t)
Explanation: Y(s)=2/[s³(s+2)]; partial fractions → stated form.
Q523. Solve: y''(t)+y(t)=t², y(0)=0, y'(0)=0.
A) y(t) = t² - 2 + 2cos(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Y(s)=2/[s³(s²+1)]; inverse → t² - 2 + 2cos(t).
Q524. Solve: y''(t)-y(t)=t², y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t² ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Y(s)=2/[s³(s²-1)]; inverse → sinh(t) - t².
Q525. Solve: y''(t)+y(t)=t³, y(0)=0, y'(0)=0.
A) y(t) = t³ - 6t + 6sin(t) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Y(s)=6/[s⁴(s²+1)]; inverse → t³ - 6t + 6sin(t).
Q526. Solve: y''(t)-y(t)=t³, y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t³ ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Y(s)=6/[s⁴(s²-1)]; inverse → sinh(t) - t³.
Q527. Solve: y'(t)+2y(t)=t³, y(0)=0.
A) y(t) = t³ - 3t² + 6t - 6 + 6e^{-2t} ✅
B) y(t) = t³
C) y(t) = e^{-2t}
D) y(t) = sin(t)
Explanation: Y(s)=6/[s⁴(s+2)]; inverse → polynomial minus exponential.
Q528. Solve: y''(t)+y(t)=t⁴, y(0)=0, y'(0)=0.
A) y(t) = t⁴ - 12t² + 24 - 24cos(t) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Y(s)=24/[s⁵(s²+1)]; inverse → polynomial with cosine term.
Q529. Solve: y''(t)-y(t)=t⁴, y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t⁴ ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Y(s)=24/[s⁵(s²-1)]; inverse → sinh(t) - t⁴.
Q530. Solve: y'(t)+y(t)=t⁴, y(0)=0.
A) y(t) = t⁴ - 4t³ + 12t² - 24t + 24 - 24e^{-t} ✅
B) y(t) = t⁴
C) y(t) = e^{-t}
D) y(t) = sin(t)
Explanation: Y(s)=24/[s⁵(s+1)]; inverse → polynomial minus exponential.
Q531. Solve: y''(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = sin(t) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Impulse input; transfer function 1/(s²+1); output sin(t).
Q532. Solve: y''(t)+2y'(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = e^{-t} ✅
B) y(t) = e^t
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Impulse response of system with (s+1)² denominator → e^{-t}.
Q533. Solve: y''(t)+4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sin(2t) ✅
B) y(t) = cos(2t)
C) y(t) = e^{2t}
D) y(t) = sinh(2t)
Explanation: Transfer function 1/(s²+4); inverse → (1/2)sin(2t).
Q534. Solve: y''(t)-4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sinh(2t) ✅
B) y(t) = cosh(2t)
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s²-4); inverse → (1/2)sinh(2t).
Q535. Solve: y'(t)+2y(t)=δ(t), y(0)=0.
A) y(t) = e^{-2t} ✅
B) y(t) = e^{2t}
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s+2); inverse → e^{-2t}.
Q536. Solve: y'(t)-2y(t)=δ(t), y(0)=0.
A) y(t) = e^{2t} ✅
B) y(t) = e^{-2t}
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s-2); inverse → e^{2t}.
Q537. Solve: y''(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = 1 - cos(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Step input; transfer function 1/(s²+1); inverse → 1 - cos(t).
Q538. Solve: y''(t)+2y'(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = t e^{-t} ✅
B) y(t) = e^{-t}
C) y(t) = 1 - e^{-t}
D) y(t) = sin(t)
Explanation: Step input; transfer function 1/(s+1)²; inverse → t e^{-t}.
Q539. Solve: y''(t)+4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1 - cos(2t)) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^{2t}
Explanation: Step input; transfer function 1/(s²+4); inverse → (1/2)(1 - cos(2t)).
Q540. Solve: y''(t)-4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(cosh(2t) - 1) ✅
B) y(t) = sinh(2t)
C) y(t) = cosh(2t)
D) y(t) = e^{2t}
Explanation: Step input; transfer function 1/(s²-4); inverse → (1/4)(cosh(2t)-1).
Q541. Solve: y'(t)+y(t)=u(t), y(0)=0.
A) y(t) = 1 - e^{-t} ✅
B) y(t) = e^{-t}
C) y(t) = t
D) y(t) = sin(t)
Explanation: Step input; transfer function 1/(s+1); inverse → 1 - e^{-t}.
Q542. Solve: y'(t)-y(t)=u(t), y(0)=0.
A) y(t) = e^t - 1 ✅
B) y(t) = e^{-t}
C) y(t) = t
D) y(t) = sin(t)
Explanation: Step input; transfer function 1/(s-1); inverse → e^t - 1.
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Q543. Solve: y''(t)+y(t)=t³, y(0)=0, y'(0)=0.
A) y(t) = t³ - 6t + 6sin(t) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Y(s)=6/[s⁴(s²+1)]; inverse → polynomial + sinusoidal term.
Q544. Solve: y''(t)-y(t)=t³, y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t³ ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Y(s)=6/[s⁴(s²-1)]; inverse → hyperbolic minus polynomial.
Q545. Solve: y'(t)+2y(t)=t³, y(0)=0.
A) y(t) = t³ - 3t² + 6t - 6 + 6e^{-2t} ✅
B) y(t) = t³
C) y(t) = e^{-2t}
D) y(t) = sin(t)
Explanation: Y(s)=6/[s⁴(s+2)]; inverse → polynomial minus exponential.
Q546. Solve: y''(t)+y(t)=t⁴, y(0)=0, y'(0)=0.
A) y(t) = t⁴ - 12t² + 24 - 24cos(t) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Y(s)=24/[s⁵(s²+1)]; inverse → polynomial with cosine correction.
Q547. Solve: y''(t)-y(t)=t⁴, y(0)=0, y'(0)=0.
A) y(t) = sinh(t) - t⁴ ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Y(s)=24/[s⁵(s²-1)]; inverse → hyperbolic minus polynomial.
Q548. Solve: y'(t)+y(t)=t⁴, y(0)=0.
A) y(t) = t⁴ - 4t³ + 12t² - 24t + 24 - 24e^{-t} ✅
B) y(t) = t⁴
C) y(t) = e^{-t}
D) y(t) = sin(t)
Explanation: Y(s)=24/[s⁵(s+1)]; inverse → polynomial minus exponential.
Q549. Solve: y''(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = sin(t) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Impulse input; transfer function 1/(s²+1); output sin(t).
Q550. Solve: y''(t)+2y'(t)+y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = e^{-t} ✅
B) y(t) = e^t
C) y(t) = cos(t)
D) y(t) = sin(t)
Explanation: Impulse response of system with (s+1)² denominator → e^{-t}.
Q551. Solve: y''(t)+4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sin(2t) ✅
B) y(t) = cos(2t)
C) y(t) = e^{2t}
D) y(t) = sinh(2t)
Explanation: Transfer function 1/(s²+4); inverse → (1/2)sin(2t).
Q552. Solve: y''(t)-4y(t)=δ(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sinh(2t) ✅
B) y(t) = cosh(2t)
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s²-4); inverse → (1/2)sinh(2t).
Q553. Solve: y'(t)+2y(t)=δ(t), y(0)=0.
A) y(t) = e^{-2t} ✅
B) y(t) = e^{2t}
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s+2); inverse → e^{-2t}.
Q554. Solve: y'(t)-2y(t)=δ(t), y(0)=0.
A) y(t) = e^{2t} ✅
B) y(t) = e^{-2t}
C) y(t) = cos(2t)
D) y(t) = sin(2t)
Explanation: Transfer function 1/(s-2); inverse → e^{2t}.
Q555. Solve: y''(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = 1 - cos(t) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Step input; transfer function 1/(s²+1); inverse → 1 - cos(t).
Q556. Solve: y''(t)+2y'(t)+y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = t e^{-t} ✅
B) y(t) = e^{-t}
C) y(t) = 1 - e^{-t}
D) y(t) = sin(t)
Explanation: Step input; transfer function 1/(s+1)²; inverse → t e^{-t}.
Q557. Solve: y''(t)+4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1 - cos(2t)) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^{2t}
Explanation: Step input; transfer function 1/(s²+4); inverse → (1/2)(1 - cos(2t)).
Q558. Solve: y''(t)-4y(t)=u(t), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(cosh(2t) - 1) ✅
B) y(t) = sinh(2t)
C) y(t) = cosh(2t)
D) y(t) = e^{2t}
Explanation: Step input; transfer function 1/(s²-4); inverse → (1/4)(cosh(2t)-1).
Q559. Solve: y'(t)+y(t)=u(t), y(0)=0.
A) y(t) = 1 - e^{-t} ✅
B) y(t) = e^{-t}
C) y(t) = t
D) y(t) = sin(t)
Explanation: Step input; transfer function 1/(s+1); inverse → 1 - e^{-t}.
Q560. Solve: y'(t)-y(t)=u(t), y(0)=0.
A) y(t) = e^t - 1 ✅
B) y(t) = e^{-t}
C) y(t) = t
D) y(t) = sin(t)
Explanation: Step input; transfer function 1/(s-1); inverse → e^t - 1.
Q561. For system G(s)=1/(s+1), find impulse response.
A) g(t)=e^{-t} ✅
B) g(t)=e^{t}
C) g(t)=cos(t)
D) g(t)=sin(t)
Explanation: Inverse Laplace of 1/(s+1) → e^{-t}.
Q562. For system G(s)=1/(s+2), find impulse response.
A) g(t)=e^{-2t} ✅
B) g(t)=e^{2t}
C) g(t)=cos(2t)
D) g(t)=sin(2t)
Explanation: Inverse Laplace of 1/(s+2) → e^{-2t}.
Q563. For system G(s)=1/(s-2), find impulse response.
A) g(t)=e^{2t} ✅
B) g(t)=e^{-2t}
C) g(t)=cos(2t)
D) g(t)=sin(2t)
Explanation: Inverse Laplace of 1/(s-2) → e^{2t}.
Q564. For system G(s)=1/(s²+1), find impulse response.
A) g(t)=sin(t) ✅
B) g(t)=cos(t)
C) g(t)=e^t
D) g(t)=sinh(t)
Explanation: Inverse Laplace of 1/(s²+1) → sin(t).
Q565. For system G(s)=1/(s²+4), find impulse response.
A) g(t)=(1/2)sin(2t) ✅
B) g(t)=cos(2t)
C) g(t)=e^{2t}
D) g(t)=sinh(2t)
Explanation: Inverse Laplace of 1/(s²+4) → (1/2)sin(2t).
Q566. For system G(s)=1/(s²-4), find impulse response.
A) g(t)=(1/2)sinh(2t) ✅
B) g(t)=cosh(2t)
C) g(t)=e^{2t}
D) g(t)=sin(2t)
Explanation: Inverse Laplace of 1/(s²-4) → (1/2)sinh(2t).
Q567. For system G(s)=1/(s+1), find step response.
A) g(t)=1 - e^{-t} ✅
B) g(t)=e^{-t}
C) g(t)=t
D) g(t)=sin(t)
Explanation: Step input → Laplace 1/s; output 1/[s(s+1)] → 1 - e^{-t}.
Q568. For system G(s)=1/(s+2), find step response.
A) g(t)=1 - e^{-2t} ✅
B) g(t)=e^{-2t}
C) g(t)=t
D) g(t)=sin(t)
Explanation: Step input → Laplace 1/s; output 1/[s(s+2)] → 1 - e^{-2t}.
Q569. For system G(s)=1/(s-2), find step response.
A) g(t)=e^{2t} - 1 ✅
B) g(t)=e^{-2t}
C) g(t)=t
D) g(t)=sin(t)
Explanation: Step input → Laplace 1/s; output 1/[s(s-2)] → e^{2t} - 1.
Q570. For system G(s)=1/(s²+1), find step response.
A) g(t)=1 - cos(t) ✅
B) g(t)=sin(t)
C) g(t)=cos(t)
D) g(t)=e^t
Explanation: Step input → Laplace 1/s; output 1/[s(s²+1)] → 1 - cos(t).
Q571. For system G(s)=1/(s²+4), find step response.
A) g(t)=(1/2)(1 - cos(2t)) ✅
B) g(t)=sin(2t)
C) g(t)=cos(2t)
D) g(t)=e^{2t}
Explanation: Step input → Laplace 1/s; output 1/[s(s²+4)] → (1/2)(1 - cos(2t)).
Q572. For system G(s)=1/(s²-4), find step response.
A) g(t)=(1/4)(cosh(2t) - 1) ✅
B) g(t)=sinh(2t)
C) g(t)=cosh(2t)
D) g(t)=e^{2t}
Explanation: Step input → Laplace 1/s; output 1/[s(s²-4)] → (1/4)(cosh(2t)-1).
Q573. For system G(s)=1/(s+1)², find impulse response.
A) g(t)=t e^{-t} ✅
B) g(t)=e^{-t}
C) g(t)=1 - e^{-t}
D) g(t)=sin(t)
Explanation: Inverse Laplace of 1/(s+1)² → t e^{-t}.
Q574. For system G(s)=1/(s+1)², find step response.
A) g(t)=1 - (1+t)e^{-t} ✅
B) g(t)=e^{-t}
C) g(t)=t
D) g(t)=sin(t)
Explanation: Step input → Laplace 1/s; output 1/[s(s+1)²]; inverse → 1 - (1+t)e^{-t}.
Q575. For system G(s)=1/(s+2)², find impulse response.
A) g(t)=t e^{-2t} ✅
B) g(t)=e^{-2t}
C) g(t)=1 - e^{-2t}
D) g(t)=sin(2t)
Explanation: Inverse Laplace of 1/(s+2)² → t e^{-2t}.
Q576. For system G(s)=1/(s+2)², find step response.
A) g(t)=1 - (1+2t)e^{-2t} ✅
B) g(t)=e^{-2t}
C) g(t)=t
D) g(t)=sin(2t)
Explanation: Step input → Laplace 1/s; output 1/[s(s+2)²]; inverse → 1 - (1+2t)e^{-2t}.
Q577. For system G(s)=1/(s²+1)², find impulse response.
A) g(t)=(1/2)(sin(t) - t cos(t)) ✅
B) g(t)=sin(t)
C) g(t)=cos(t)
D) g(t)=e^t
Explanation: Inverse Laplace of 1/(s²+1)² → (1/2)(sin(t) - t cos(t)).
Q578. For system G(s)=1/(s²+1)², find step response.
A) g(t)=(1/2)(1 - cos(t) - t sin(t)) ✅
B) g(t)=sin(t)
C) g(t)=cos(t)
D) g(t)=e^t
Explanation: Step input → Laplace 1/s; output 1/[s(s²+1)²]; inverse → (1/2)(1 - cos(t) - t sin(t)).
Q579. For system G(s)=1/(s²+4)², find impulse response.
A) g(t)=(1/4)(sin(2t) - 2t cos(2t)) ✅
B) g(t)=sin(2t)
C) g(t)=cos(2t)
D) g(t)=e^{2t}
Explanation: Inverse Laplace of 1/(s²+4)² → (1/4)(sin(2t) - 2t cos(2t)).
Q580. For system G(s)=1/(s²+4)², find step response.
A) g(t)=(1/4)(1 - cos(2t) - t sin(2t)) ✅
B) g(t)=sin(2t)
C) g(t)=cos(2t)
D) g(t)=e^{2t}
Explanation: Step input → Laplace 1/s; output 1/[s(s²+4)²]; inverse → (1/4)(1 - cos(2t) - t sin(2t)).
Q581. For X(s)=1/(s+1), find Fourier transform if ROC includes jω-axis.
A) X(jω)=1/(jω+1) ✅
B) X(jω)=1/(jω-1)
C) X(jω)=1/(ω²+1)
D) X(jω)=e^{-jω}
Explanation: Replace s=jω; ROC includes jω-axis → valid Fourier transform.
Q582. For X(s)=1/(s²+1), find Fourier transform.
A) X(jω)=1/(ω²+1) ✅
B) X(jω)=1/(jω+1)
C) X(jω)=1/(jω-1)
D) X(jω)=e^{-jω}
Explanation: Replace s=jω → 1/(ω²+1).
Q583. For X(s)=1/(s²+4), find Fourier transform.
A) X(jω)=1/(ω²+4) ✅
B) X(jω)=1/(jω+2)
C) X(jω)=1/(jω-2)
D) X(jω)=e^{-jω}
Explanation: Replace s=jω → 1/(ω²+4).
Q584. For X(s)=1/(s²-4), find Fourier transform if ROC includes jω-axis.
A) X(jω)=1/(ω²-4) ✅
B) X(jω)=1/(ω²+4)
C) X(jω)=1/(jω+2)
D) X(jω)=e^{-jω}
Explanation: Replace s=jω → 1/(ω²-4).
Q585. For system G(s)=1/(s+1), find frequency response.
A) G(jω)=1/(jω+1) ✅
B) G(jω)=1/(jω-1)
C) G(jω)=1/(ω²+1)
D) G(jω)=e^{-jω}
Explanation: Frequency response = Laplace with s=jω.
Q586. For system G(s)=1/(s²+1), find frequency response.
A) G(jω)=1/(ω²+1) ✅
B) G(jω)=1/(jω+1)
C) G(jω)=1/(jω-1)
D) G(jω)=e^{-jω}
Explanation: Frequency response = Laplace with s=jω.
Q587. For system G(s)=1/(s²+4), find frequency response.
A) G(jω)=1/(ω²+4) ✅
B) G(jω)=1/(jω+2)
C) G(jω)=1/(jω-2)
D) G(jω)=e^{-jω}
Explanation: Frequency response = Laplace with s=jω.
Q588. For system G(s)=1/(s²-4), find frequency response if ROC includes jω-axis.
A) G(jω)=1/(ω²-4) ✅
B) G(jω)=1/(ω²+4)
C) G(jω)=1/(jω+2)
D) G(jω)=e^{-jω}
Explanation: Frequency response = Laplace with s=jω.
Q589. For x(t)=e^{-t}u(t), find Fourier transform.
A) X(jω)=1/(1+jω) ✅
B) X(jω)=1/(1-jω)
C) X(jω)=1/(ω²+1)
D) X(jω)=e^{-jω}
Explanation: Laplace X(s)=1/(s+1); ROC Re(s)>-1; Fourier valid → 1/(1+jω).
Q590. For x(t)=e^{-2t}u(t), find Fourier transform.
A) X(jω)=1/(2+jω) ✅
B) X(jω)=1/(2-jω)
C) X(jω)=1/(ω²+4)
D) X(jω)=e^{-jω}
Explanation: Laplace X(s)=1/(s+2); ROC Re(s)>-2; Fourier valid → 1/(2+jω).
Q591. For x(t)=cos(t)u(t), find Fourier transform.
A) X(jω)=jω/(ω²-1) ✅
B) X(jω)=1/(ω²+1)
C) X(jω)=1/(jω+1)
D) X(jω)=e^{-jω}
Explanation: Laplace X(s)=s/(s²+1); replace s=jω → jω/(ω²-1).
Q592. For x(t)=sin(t)u(t), find Fourier transform.
A) X(jω)=1/(ω²-1) ✅
B) X(jω)=1/(ω²+1)
C) X(jω)=1/(jω+1)
D) X(jω)=e^{-jω}
Explanation: Laplace X(s)=1/(s²+1); replace s=jω → 1/(ω²-1).
Q593. For x(t)=cos(2t)u(t), find Fourier transform.
A) X(jω)=jω/(ω²-4) ✅
B) X(jω)=1/(ω²+4)
C) X(jω)=1/(jω+2)
D) X(jω)=e^{-jω}
Explanation: Laplace X(s)=s/(s²+4); replace s=jω → jω/(ω²-4).
Q594. For x(t)=sin(2t)u(t), find Fourier transform.
A) X(jω)=2/(ω²-4) ✅
B) X(jω)=1/(ω²+4)
C) X(jω)=1/(jω+2)
D) X(jω)=e^{-jω}
Explanation: Laplace X(s)=2/(s²+4); replace s=jω → 2/(ω²-4).
Q595. For x(t)=cosh(t)u(t), find Fourier transform.
A) X(jω)=jω/(ω²+1) ✅
B) X(jω)=1/(ω²-1)
C) X(jω)=1/(jω+1)
D) X(jω)=e^{-jω}
Explanation: Laplace X(s)=s/(s²-1); replace s=jω → jω/(ω²+1).
Q596. For x(t)=sinh(t)u(t), find Fourier transform.
A) X(jω)=1/(ω²+1) ✅
B) X(jω)=1/(ω²-1)
C) X(jω)=1/(jω+1)
D) X(jω)=e^{-jω}
Explanation: Laplace X(s)=1/(s²-1); replace s=jω → 1/(ω²+1).
Q597. For system G(s)=1/(s+1), find magnitude response.
A) |G(jω)|=1/√(ω²+1) ✅
B) |G(jω)|=1/(ω²+1)
C) |G(jω)|=1/(1+ω)
D) |G(jω)|=e^{-ω}
Explanation: G(jω)=1/(jω+1); magnitude → 1/√(ω²+1).
Q598. For system G(s)=1/(s+2), find magnitude response.
A) |G(jω)|=1/√(ω²+4) ✅
B) |G(jω)|=1/(ω²+4)
C) |G(jω)|=1/(2+ω)
D) |G(jω)|=e^{-ω}
Explanation: G(jω)=1/(jω+2); magnitude → 1/√(ω²+4).
Q599. For system G(s)=1/(s²+1), find magnitude response.
A) |G(jω)|=1/(ω²+1) ✅
B) |G(jω)|=1/√(ω²+1)
C) |G(jω)|=1/(1+ω)
D) |G(jω)|=e^{-ω}
Explanation: G(jω)=1/(ω²+1); magnitude → 1/(ω²+1).
Q600. For system G(s)=1/(s²+4), find magnitude response.
A) |G(jω)|=1/(ω²+4) ✅
B) |G(jω)|=1/√(ω²+4)
C) |G(jω)|=1/(2+ω)
D) |G(jω)|=e^{-ω}
Explanation: G(jω)=1/(ω²+4); magnitude → 1/(ω²+4).
Q601. For RLC circuit: V(s)=1/s, I(s)=V(s)/(s+1), find i(t).
A) i(t)=1 - e^{-t} ✅
B) i(t)=e^{-t}
C) i(t)=sin(t)
D) i(t)=cos(t)
Explanation: Step input; transfer 1/(s+1); inverse → 1 - e^{-t}.
Q602. For RL circuit: V(s)=1/s, I(s)=V(s)/(s+2), find i(t).
A) i(t)=1 - e^{-2t} ✅
B) i(t)=e^{-2t}
C) i(t)=sin(2t)
D) i(t)=cos(2t)
Explanation: Step input; transfer 1/(s+2); inverse → 1 - e^{-2t}.
Q603. For RC circuit: V(s)=1/s, I(s)=V(s)/(s+1), find i(t).
A) i(t)=1 - e^{-t} ✅
B) i(t)=e^{-t}
C) i(t)=sin(t)
D) i(t)=cos(t)
Explanation: Same as RL with Ï„=1; inverse → 1 - e^{-t}.
Q604. For LC circuit: V(s)=1/s, I(s)=V(s)/(s²+1), find i(t).
A) i(t)=1 - cos(t) ✅
B) i(t)=sin(t)
C) i(t)=cos(t)
D) i(t)=e^t
Explanation: Step input; transfer 1/(s²+1); inverse → 1 - cos(t).
Q605. For mass-spring system: F(s)=1/s, X(s)=F(s)/(s²+1), find x(t).
A) x(t)=1 - cos(t) ✅
B) x(t)=sin(t)
C) x(t)=cos(t)
D) x(t)=e^t
Explanation: Step input; transfer 1/(s²+1); inverse → 1 - cos(t).
Q606. For mass-spring-damper: F(s)=1/s, X(s)=F(s)/(s²+2s+1), find x(t).
A) x(t)=1 - (1+t)e^{-t} ✅
B) x(t)=e^{-t}
C) x(t)=sin(t)
D) x(t)=cos(t)
Explanation: Step input; transfer 1/(s+1)²; inverse → 1 - (1+t)e^{-t}.
Q607. For probability: Laplace transform of f(t)=e^{-t}, t≥0.
A) F(s)=1/(s+1) ✅
B) F(s)=1/(s-1)
C) F(s)=1/(s²+1)
D) F(s)=e^{-s}
Explanation: ∫₀∞ e^{-t}e^{-st}dt=1/(s+1).
Q608. For probability: Laplace transform of f(t)=t e^{-t}, t≥0.
A) F(s)=1/(s+1)² ✅
B) F(s)=1/(s-1)²
C) F(s)=1/(s²+1)
D) F(s)=e^{-s}
Explanation: ∫₀∞ t e^{-(s+1)t}dt=1/(s+1)².
Q609. For probability: Laplace transform of f(t)=t² e^{-t}, t≥0.
A) F(s)=2/(s+1)³ ✅
B) F(s)=2/(s-1)³
C) F(s)=1/(s²+1)
D) F(s)=e^{-s}
Explanation: ∫₀∞ t² e^{-(s+1)t}dt=2/(s+1)³.
Q610. For PDE: u_t=u_xx, initial δ(x).
A) U(s)=e^{-√s |x|} ✅
B) U(s)=e^{-s|x|}
C) U(s)=e^{-x}
D) U(s)=cos(x)
Explanation: Heat equation fundamental solution; Laplace in time → exponential in √s.
Q611. For PDE: u_t+u_x=0, initial δ(x).
A) U(s)=e^{-sx} ✅
B) U(s)=e^{-√s x}
C) U(s)=e^{-x}
D) U(s)=cos(x)
Explanation: Transport equation; Laplace in time → exponential in s.
Q612. For PDE: u_t+u=0, initial u(0)=1.
A) U(s)=1/(s+1) ✅
B) U(s)=1/(s-1)
C) U(s)=1/s
D) U(s)=e^{-s}
Explanation: ODE in time; Laplace → 1/(s+1).
Q613. For PDE: u_t+2u=0, initial u(0)=1.
A) U(s)=1/(s+2) ✅
B) U(s)=1/(s-2)
C) U(s)=1/s
D) U(s)=e^{-s}
Explanation: ODE in time; Laplace → 1/(s+2).
Q614. For PDE: u_t=u, initial u(0)=1.
A) U(s)=1/(s-1) ✅
B) U(s)=1/(s+1)
C) U(s)=1/s
D) U(s)=e^{-s}
Explanation: ODE in time; Laplace → 1/(s-1).
Q615. For PDE: u_t=2u, initial u(0)=1.
A) U(s)=1/(s-2) ✅
B) U(s)=1/(s+2)
C) U(s)=1/s
D) U(s)=e^{-s}
Explanation: ODE in time; Laplace → 1/(s-2).
Q616. For PDE: u_t=u_x, initial δ(x).
A) U(s)=δ(s-x) ✅
B) U(s)=e^{-sx}
C) U(s)=e^{-√s x}
D) U(s)=cos(x)
Explanation: Transport equation; Laplace in time → δ(s-x).
Q617. For PDE: u_t=u_xx, initial u(x,0)=1.
A) U(s)=1/s ✅
B) U(s)=1/(s+1)
C) U(s)=1/(s-1)
D) U(s)=e^{-s}
Explanation: Heat equation with constant initial condition → Laplace 1/s.
Q618. For PDE: u_t+u_xx=0, initial δ(x).
A) U(s)=1/(s+ω²) ✅
B) U(s)=1/(s-ω²)
C) U(s)=1/s
D) U(s)=e^{-s}
Explanation: Diffusion equation; Laplace → 1/(s+ω²).
Q619. For PDE: u_t+u_xx=0, initial u(x,0)=1.
A) U(s)=1/s ✅
B) U(s)=1/(s+1)
C) U(s)=1/(s-1)
D) U(s)=e^{-s}
Explanation: Constant initial condition → Laplace 1/s.
Q620. For PDE: u_t+u_x=0, initial u(x,0)=1.
A) U(s)=1/s ✅
B) U(s)=1/(s+1)
C) U(s)=1/(s-1)
D) U(s)=e^{-s}
Explanation: Transport equation with constant initial condition → Laplace 1/s.
Q621. Solve: y''(t)+y(t)=u(t-Ï€), y(0)=0, y'(0)=0.
A) y(t) = (1 - cos(t-Ï€)) u(t-Ï€) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Shifted step; transfer 1/(s²+1); inverse → (1 - cos(t-Ï€))u(t-Ï€).
Q622. Solve: y''(t)+4y(t)=δ(t-π), y(0)=0, y'(0)=0.
A) y(t) = (1/2) sin(2(t-Ï€)) u(t-Ï€) ✅
B) y(t) = cos(2t)
C) y(t) = e^{2t}
D) y(t) = sinh(2t)
Explanation: Shifted impulse; transfer 1/(s²+4); inverse → (1/2)sin(2(t-Ï€))u(t-Ï€).
Q623. Solve: y''(t)-4y(t)=u(t-Ï€), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(cosh(2(t-Ï€)) - 1) u(t-Ï€) ✅
B) y(t) = sinh(2t)
C) y(t) = cosh(2t)
D) y(t) = e^{2t}
Explanation: Shifted step; transfer 1/(s²-4); inverse → (1/4)(cosh(2(t-Ï€))-1)u(t-Ï€).
Q624. Solve: y'(t)+2y(t)=δ(t-π), y(0)=0.
A) y(t) = e^{-2(t-Ï€)} u(t-Ï€) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted impulse; transfer 1/(s+2); inverse → e^{-2(t-Ï€)}u(t-Ï€).
Q625. Solve: y'(t)-2y(t)=u(t-Ï€), y(0)=0.
A) y(t) = (e^{2(t-Ï€)} - 1) u(t-Ï€) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Shifted step; transfer 1/(s-2); inverse → (e^{2(t-Ï€)} - 1)u(t-Ï€).
Q626. Solve: y''(t)+y(t)=cos(t)u(t-Ï€), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1 - cos(t-Ï€) - (t-Ï€) sin(t-Ï€)) u(t-Ï€) ✅
B) y(t) = cos(t)
C) y(t) = sin(t)
D) y(t) = e^t
Explanation: Shifted cosine forcing; partial fractions → stated form.
Q627. Solve: y''(t)+4y(t)=sin(2t)u(t-Ï€), y(0)=0, y'(0)=0.
A) y(t) = (1/4)(sin(2(t-Ï€)) - 2(t-Ï€) cos(2(t-Ï€))) u(t-Ï€) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^{2t}
Explanation: Resonant forcing with shift; inverse → polynomial–trig mix.
Q628. Solve: y'(t)+y(t)=t u(t-Ï€), y(0)=0.
A) y(t) = (t-Ï€ - 1 + e^{-(t-Ï€)}) u(t-Ï€) ✅
B) y(t) = t
C) y(t) = e^{-t}
D) y(t) = sin(t)
Explanation: Shifted polynomial forcing; inverse → polynomial minus exponential.
Q629. Solve: y''(t)+y(t)=t² u(t-Ï€), y(0)=0, y'(0)=0.
A) y(t) = (t-Ï€)² - 2 + 2cos(t-Ï€) u(t-Ï€) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Shifted polynomial forcing; inverse → polynomial + cosine.
Q630. Solve: y''(t)-y(t)=t³ u(t-Ï€), y(0)=0, y'(0)=0.
A) y(t) = (sinh(t-Ï€) - (t-Ï€)³) u(t-Ï€) ✅
B) y(t) = cosh(t)
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Shifted polynomial forcing; inverse → hyperbolic minus polynomial.
Q631. Solve: y'(t)+2y(t)=t² u(t-Ï€), y(0)=0.
A) y(t) = ((t-Ï€)² - (t-Ï€) + 1/2 - (1/2)e^{-2(t-Ï€)}) u(t-Ï€) ✅
B) y(t) = t²
C) y(t) = e^{-2t}
D) y(t) = sin(t)
Explanation: Shifted polynomial forcing; inverse → polynomial minus exponential.
Q632. Solve: y''(t)+y(t)=δ(t-π)+δ(t-2π), y(0)=0, y'(0)=0.
A) y(t) = sin(t-Ï€)u(t-Ï€) + sin(t-2Ï€)u(t-2Ï€) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Multiple impulses; superposition principle → sum of shifted sines.
Q633. Solve: y''(t)+4y(t)=u(t-Ï€)+u(t-2Ï€), y(0)=0, y'(0)=0.
A) y(t) = (1/2)(1 - cos(2(t-Ï€)))u(t-Ï€) + (1/2)(1 - cos(2(t-2Ï€)))u(t-2Ï€) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^{2t}
Explanation: Multiple steps; superposition principle → sum of shifted responses.
Q634. Solve: y'(t)+y(t)=δ(t-π)+δ(t-2π), y(0)=0.
A) y(t) = e^{-(t-Ï€)}u(t-Ï€) + e^{-(t-2Ï€)}u(t-2Ï€) ✅
B) y(t) = e^{-t}
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Multiple impulses; superposition principle → sum of exponentials.
Q635. Solve: y'(t)+2y(t)=u(t-Ï€)+u(t-2Ï€), y(0)=0.
A) y(t) = (1 - e^{-2(t-Ï€)})u(t-Ï€) + (1 - e^{-2(t-2Ï€)})u(t-2Ï€) ✅
B) y(t) = e^{-2t}
C) y(t) = e^{2t}
D) y(t) = sin(2t)
Explanation: Multiple steps; superposition principle → sum of exponentials.
Q636. Solve: y''(t)+y(t)=periodic δ(t-nπ), y(0)=0, y'(0)=0.
A) y(t) = Σ sin(t-nÏ€)u(t-nÏ€) ✅
B) y(t) = cos(t)
C) y(t) = e^t
D) y(t) = sinh(t)
Explanation: Periodic impulses; solution is infinite sum of shifted sines.
Q637. Solve: y''(t)+4y(t)=periodic u(t-nπ), y(0)=0, y'(0)=0.
A) y(t) = Σ (1/2)(1 - cos(2(t-nÏ€)))u(t-nÏ€) ✅
B) y(t) = sin(2t)
C) y(t) = cos(2t)
D) y(t) = e^{2t}
Explanation: Periodic steps; solution is infinite sum of shifted responses.
Q638. Solve: y'(t)+y(t)=periodic δ(t-nπ), y(0)=0.
A) y(t) = Σ e^{-(t-nÏ€)} u(t-nÏ€) ✅
B) y(t) = e^{-t}
C) y(t) = e^t
D) y(t) = sin(t)
Explanation: Periodic impulses; solution is infinite sum of shifted exponentials.
Q639. Solve: y''(t)+y(t)=periodic u(t-nπ), y(0)=0, y'(0)=0.
A) y(t) = Σ (1 - cos(t-nÏ€)) u(t-nÏ€) ✅
B) y(t) = sin(t)
C) y(t) = cos(t)
D) y(t) = e^t
Explanation: Periodic steps; solution is infinite sum of shifted cosine responses.
Q640. Solve: y''(t)+4y(t)=periodic δ(t-nπ), y(0)=0, y'(0)=0.
A) y(t) = Σ (1/2) sin(2(t-nÏ€)) u(t-nÏ€) ✅
B) y(t) = cos(2t)
C) y(t) = e^{2t}
D) y(t) = sinh(2t)
Explanation: Periodic impulses; solution is infinite sum of shifted sine responses.
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