Application of Laplace Transform in solving Linear differential equations MCQs With Answer

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Introduction: Application of Laplace Transform in solving Linear differential equations MCQs With Answer is an essential topic for B.Pharm students who need firm mathematical tools for pharmacokinetics and drug process modeling. The Laplace transform simplifies solving linear ordinary differential equations (ODEs) with initial conditions, step inputs, impulses and sinusoidal forcing often seen in compartmental models, absorption–elimination kinetics and bioavailability studies. This concise, keyword-rich guide focuses on transform properties, inverse transforms, convolution, shifting theorems, and application to one- and two-compartment models. Clear conceptual questions and practice problems build familiarity with transfer functions and initial/final value theorems. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the Laplace transform of f(t)=e^{at}?

  • 1/(s-a)
  • 1/(s+a)
  • s/(s-a)
  • a/(s-a)

Correct Answer: 1/(s-a)

Q2. The Laplace transform of f'(t) assuming zero initial condition is?

  • sF(s)
  • F(s)/s
  • s^2 F(s)
  • F'(s)

Correct Answer: sF(s)

Q3. For an initial value problem y'(t)+2y(t)=e^{-t}, y(0)=1, which algebraic equation in Laplace domain is correct?

  • (s+2)Y(s)=1+s/(s+1)
  • (s+2)Y(s)=1+1/(s+1)
  • (s+2)Y(s)=1+1/(s-1)
  • (s+2)Y(s)=s+1/(s+1)

Correct Answer: (s+2)Y(s)=1+1/(s+1)

Q4. What is the inverse Laplace of 1/(s^2 + ω^2)?

  • cos(ωt)
  • sin(ωt)
  • (1/ω)sin(ωt)
  • (1/ω)cos(ωt)

Correct Answer: (1/ω)sin(ωt)

Q5. The first shifting theorem (time shift) for Laplace transforms states that L{f(t-a)u(t-a)} equals?

  • e^{as}F(s)
  • e^{-as}F(s)
  • F(s-a)
  • F(s)/e^{as}

Correct Answer: e^{-as}F(s)

Q6. In pharmacokinetics, solving linear ODEs with Laplace transform helps to model which process?

  • Drug absorption and elimination
  • DNA sequencing
  • Microscopic imaging
  • Bacterial culture growth only

Correct Answer: Drug absorption and elimination

Q7. Laplace transform of δ(t-a) (Dirac impulse at t=a) is:

  • e^{-as}
  • e^{as}
  • 1
  • δ(s-a)

Correct Answer: e^{-as}

Q8. Which property is used to convert convolution in time domain to multiplication in s-domain?

  • Convolution theorem
  • Linearity
  • Time shifting
  • Frequency scaling

Correct Answer: Convolution theorem

Q9. The Laplace transform of t^n (n a nonnegative integer) is:

  • n!/s^{n+1}
  • s^{n}/n!
  • n!/s^{n-1}
  • 1/s^{n}

Correct Answer: n!/s^{n+1}

Q10. For a linear ODE with constant coefficients, Laplace transform converts differential equation into:

  • An algebraic equation in s
  • A partial differential equation
  • A next-order differential equation
  • An integral equation in time

Correct Answer: An algebraic equation in s

Q11. Initial value theorem states lim_{t→0+} f(t) equals:

  • lim_{s→∞} sF(s)
  • lim_{s→0} sF(s)
  • lim_{s→∞} F(s)
  • lim_{s→0} F(s)

Correct Answer: lim_{s→∞} sF(s)

Q12. Final value theorem gives lim_{t→∞} f(t) as:

  • lim_{s→0} sF(s), if poles in left half-plane
  • lim_{s→∞} sF(s)
  • lim_{s→0} F(s)
  • lim_{s→∞} F(s)

Correct Answer: lim_{s→0} sF(s), if poles in left half-plane

Q13. Which partial fraction decomposition is most useful to invert rational Laplace expressions?

  • Decompose into simple fractions with linear/quadratic denominators
  • Decompose into exponentials only
  • Decompose into polynomials of t
  • Decompose into delta functions

Correct Answer: Decompose into simple fractions with linear/quadratic denominators

Q14. The Laplace transform of cos(ωt) is:

  • s/(s^2+ω^2)
  • ω/(s^2+ω^2)
  • 1/(s^2+ω^2)
  • s^2/(s^2+ω^2)

Correct Answer: s/(s^2+ω^2)

Q15. To solve y”+5y’+6y=0 using Laplace, what is the denominator of Y(s) after applying transforms and initial conditions?

  • s^2+5s+6
  • (s+2)(s+3)
  • s^2-5s+6
  • s^2+6s+5

Correct Answer: s^2+5s+6

Q16. If transfer function H(s)=1/(s+k) relates input to output, the time response to unit step is:

  • (1/k)(1-e^{-kt})
  • e^{-kt}
  • k e^{-kt}
  • 1-e^{-kt}

Correct Answer: (1/k)(1-e^{-kt})

Q17. Which Laplace pair is useful for modeling first-order elimination in pharmacokinetics?

  • L{e^{-kt}} = 1/(s+k)
  • L{e^{kt}} = 1/(s-k)
  • L{t e^{-kt}} = 1/(s+k)^2
  • L{sin(kt)} = k/(s^2+k^2)

Correct Answer: L{e^{-kt}} = 1/(s+k)

Q18. The derivative property L{f”(t)} equals:

  • s^2 F(s) – sf(0) – f'(0)
  • s F(s) – f(0)
  • F(s)/s^2
  • s^2 F(s) – f(0)

Correct Answer: s^2 F(s) – sf(0) – f'(0)

Q19. Inverse Laplace of 1/(s(s+a)) is:

  • (1/a)(1-e^{-at})
  • (1/a)e^{-at}
  • e^{-at}
  • t e^{-at}

Correct Answer: (1/a)(1-e^{-at})

Q20. The convolution f*g in time domain corresponds to what in s-domain?

  • F(s)G(s)
  • F(s)+G(s)
  • F(s)/G(s)
  • Derivative of F(s)

Correct Answer: F(s)G(s)

Q21. Which transform is most helpful to include impulse dosing in a drug model?

  • Laplace transform of delta functions
  • Fourier transform
  • Z-transform
  • Wavelet transform

Correct Answer: Laplace transform of delta functions

Q22. Solving y’+3y=sin(t), y(0)=0 with Laplace requires transform of sin(t) which is:

  • 1/(s^2+1)
  • s/(s^2+1)
  • 1/s
  • t/(s^2+1)

Correct Answer: 1/(s^2+1)

Q23. For repeated real root r of characteristic equation, inverse Laplace terms include:

  • Polynomials in t times e^{rt}
  • Sine and cosine only
  • Pure exponentials only
  • Delta functions

Correct Answer: Polynomials in t times e^{rt}

Q24. If Y(s)= (s+1)/(s^2+4s+5), the time-domain response will include:

  • e^{-2t}cos(t) and e^{-2t}sin(t) terms
  • Pure exponential e^{-t}
  • Polynomial t^2 terms
  • Delta pulses

Correct Answer: e^{-2t}cos(t) and e^{-2t}sin(t) terms

Q25. Which method is commonly combined with Laplace transforms to invert complex rational functions?

  • Partial fraction decomposition
  • Laplace differentiation with respect to s only
  • Numerical integration in time
  • Z-transform mapping

Correct Answer: Partial fraction decomposition

Q26. The Laplace transform of u(t-a) (unit step shifted) is:

  • e^{-as}/s
  • 1/s
  • e^{as}/s
  • s e^{-as}

Correct Answer: e^{-as}/s

Q27. Inverse Laplace of (s+2)/(s^2+4s+13) yields time function proportional to:

  • e^{-2t}cos(3t) + e^{-2t}sin(3t)
  • e^{2t}cos(3t)
  • e^{-4t}
  • sin(2t)

Correct Answer: e^{-2t}cos(3t) + e^{-2t}sin(3t)

Q28. When using Laplace for linear systems, poles of transfer function determine:

  • System stability and time response
  • Only the frequency response
  • Magnitude of input forcing only
  • Initial concentration only

Correct Answer: System stability and time response

Q29. The Laplace transform is linear. Therefore L{af+bg} equals:

  • aF(s)+bG(s)
  • F(s)G(s)
  • aF(s)·bG(s)
  • F(s)+G(s)+ab

Correct Answer: aF(s)+bG(s)

Q30. For ODE a y” + b y’ + c y = r(t), Laplace method requires initial conditions for:

  • y(0) and y'(0)
  • Only y(0)
  • Only y”(0)
  • No initial conditions

Correct Answer: y(0) and y'(0)

Q31. Which Laplace transform pair helps solve forced oscillatory inputs like sin(bt)?

  • L{sin(bt)} = b/(s^2+b^2)
  • L{cos(bt)} = b/(s^2+b^2)
  • L{sin(bt)} = s/(s^2+b^2)
  • L{cos(bt)} = 1/(s^2+b^2)

Correct Answer: L{sin(bt)} = b/(s^2+b^2)

Q32. To include a bolus dose at t=0 in a PK model, which time-domain input is appropriate?

  • δ(t) (Dirac delta)
  • u(t) (unit step)
  • sin(t)
  • t

Correct Answer: δ(t) (Dirac delta)

Q33. The transform L{e^{-at}f(t)} equals:

  • F(s+a)
  • F(s-a)
  • e^{-as}F(s)
  • F(s)/e^{as}

Correct Answer: F(s+a)

Q34. When solving non-homogeneous linear ODEs, the particular solution in Laplace domain is found by:

  • Algebraically solving for Y(s) and inverting
  • Guessing by inspection only
  • Numerical time stepping only
  • Using Fourier series

Correct Answer: Algebraically solving for Y(s) and inverting

Q35. Which is a correct Laplace transform of integral ∫_0^t f(τ)dτ ?

  • F(s)/s
  • sF(s)
  • F(s)·s
  • F(s)^2

Correct Answer: F(s)/s

Q36. Using Laplace transform, the response of a two-compartment model leads to:

  • Sum of two exponentials with different rates
  • Single exponential only
  • Pure sinusoidal response
  • Polynomial growth

Correct Answer: Sum of two exponentials with different rates

Q37. If Y(s)=1/(s+2)^2, the inverse Laplace transform is:

  • t e^{-2t}
  • e^{-2t}
  • (1/2)e^{-2t}
  • sin(2t)

Correct Answer: t e^{-2t}

Q38. The Laplace transform is particularly powerful compared to direct integration because it:

  • Converts differential operations into algebraic ones
  • Always gives closed-form time solutions without inversion
  • Works only for periodic inputs
  • Does not require initial conditions

Correct Answer: Converts differential operations into algebraic ones

Q39. Inverse Laplace of (s+1)/(s^2+2s+5) results in e^{-t} times:

  • cos(2t) + (1/2)sin(2t)
  • sin(2t) only
  • cos(2t) only
  • e^{2t}

Correct Answer: cos(2t) + (1/2)sin(2t)

Q40. For a linear system, if any pole has positive real part, the time response will:

  • Grow unbounded (unstable)
  • Decay to zero
  • Oscillate with constant amplitude
  • Be unaffected by poles

Correct Answer: Grow unbounded (unstable)

Q41. The Laplace transform of f(t)u(t) where u is unit step equals:

  • F(s) (assuming f defined for t≥0)
  • 0
  • 1/F(s)
  • Derivative of F(s)

Correct Answer: F(s) (assuming f defined for t≥0)

Q42. Which technique helps invert Laplace transforms with quadratic irreducible denominators?

  • Complete the square and use sine/cosine pairs
  • Use polynomial long division only
  • Ignore the quadratic
  • Use delta functions

Correct Answer: Complete the square and use sine/cosine pairs

Q43. For a first-order linear system with input u(t), solution via Laplace can be expressed as:

  • y(t)=h(t)*u(t) (convolution with impulse response)
  • y(t)=u(t)/h(t)
  • y(t)=h(t)+u(t)
  • y(t)=h(t)·u(t) (multiplication in time)

Correct Answer: y(t)=h(t)*u(t) (convolution with impulse response)

Q44. Which statement about the region of convergence (ROC) is true for Laplace transforms?

  • ROC determines convergence and stability; poles lie outside ROC
  • ROC is irrelevant for solving ODEs
  • ROC equals all complex plane always
  • ROC contains only poles

Correct Answer: ROC determines convergence and stability; poles lie outside ROC

Q45. The Laplace transform approach simplifies handling piecewise or delayed dosing using:

  • Time-shifting and unit step functions
  • Converting to Fourier series
  • Using numerical differencing only
  • Ignoring delays

Correct Answer: Time-shifting and unit step functions

Q46. When initial conditions are nonzero, Laplace transform of derivative includes:

  • Terms involving initial conditions like sf(0)
  • No dependence on initial conditions
  • Only final values
  • Division by initial condition

Correct Answer: Terms involving initial conditions like sf(0)

Q47. The transfer function for one-compartment IV bolus elimination with rate k is:

  • 1/(s+k)
  • k/(s+1)
  • s/(s+k)
  • 1/s

Correct Answer: 1/(s+k)

Q48. Which inverse Laplace gives rise to a ramp function t·u(t)?

  • 1/s^2
  • 1/s
  • s
  • e^{-s}

Correct Answer: 1/s^2

Q49. The Laplace transform aids in pharmacokinetic parameter estimation by:

  • Providing closed-form solutions for concentration vs time
  • Eliminating the need for experiments
  • Only giving frequency domain plots
  • Replacing statistical methods entirely

Correct Answer: Providing closed-form solutions for concentration vs time

Q50. When inverting Y(s) and obtaining e^{-at}u(t-a) terms, this represents:

  • A delayed exponential response starting at t=a
  • An instantaneous impulse at t=0
  • A periodic steady-state
  • A polynomial growth from zero

Correct Answer: A delayed exponential response starting at t=a

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