Laplace Transforms Questions and Answers

Q: Suppose that the function \(y(t)\) satisfies the differential equation :\[ y'' - 4y' + 4y = 0 \]with initial condition \(y(0) = 0\), \(y'(0) = 3\). Then the Laplace transform of \(y(t)\) is :

The transform is \(3/(s - 2)^{2}\).Transform the equation with \(Y = L\{y\}\), \(y(0) = 0\), \(y'(0) = 3\):\[ \begin{aligned} (s^{2}Y - 3) - 4sY + 4Y &= 0 \\ Y(s^{2} - 4s + 4) &= 3 \\ Y &= \frac{3}{(s - 2)^{2}} \end{aligned} \]So \(y = 3te^{2t}\), which satisfies both initial conditions. The \((s + 2)^{2}\) options come from a sign slip in the characteristic polynomial, and 3.5 has no source in the data.

Q: If\[ f(t) = \begin{cases} e^{-xt}g(t), & t \gt 0 \\ 0, & t \lt 0. \end{cases} \]Then which of the following is true ?

The true relation is F{f(t)} = L{g(t)}, with the transforms evaluated at matching arguments.Taking the Fourier transform of \(f\) with kernel \(e^{-iyt}\):\[ \begin{aligned} \mathcal{F}\{f\}(y) &= \int_{0}^{\infty} e^{-iyt}e^{-xt}g(t)\, dt \\ &= \int_{0}^{\infty} e^{-(x + iy)t}g(t)\, dt = \mathcal{L}\{g\}(x + iy) \end{aligned} \]So the Fourier transform of \(f\) is the Laplace transform of \(g\) at \(s = x + iy\); this is how the Laplace transform is obtained from the Fourier transform. No factor \(e^{s}\) or \(s\) appears, which rules out A and B, and D has the two transforms the wrong way round.

11.

Suppose that the function \(y(t)\) satisfies the differential equation :

\[ y'' - 4y' + 4y = 0 \]

with initial condition \(y(0) = 0\), \(y'(0) = 3\). Then the Laplace transform of \(y(t)\) is :

A.\(\dfrac{3}{(s + 2)^{2}}\)
B.\(\dfrac{3}{(s - 2)^{2}}\)
C.\(\dfrac{3.5}{(s + 2)^{2}}\)
D.\(\dfrac{3.5}{(s - 2)^{2}}\)

Exam Year(s)

MH SET - 2019, SET Exam - 2019,

Answer: B

The transform is \(3/(s - 2)^{2}\).

Transform the equation with \(Y = L\{y\}\), \(y(0) = 0\), \(y'(0) = 3\):

\[ \begin{aligned} (s^{2}Y - 3) - 4sY + 4Y &= 0 \\ Y(s^{2} - 4s + 4) &= 3 \\ Y &= \frac{3}{(s - 2)^{2}} \end{aligned} \]

So \(y = 3te^{2t}\), which satisfies both initial conditions. The \((s + 2)^{2}\) options come from a sign slip in the characteristic polynomial, and 3.5 has no source in the data.

Discuss in Forum

12.

\[ f(t) = \begin{cases} e^{-xt}g(t), & t \gt 0 \\ 0, & t \lt 0. \end{cases} \]

Then which of the following is true ?

A.\(L\{f(t)\} = e^{s}F\{g(t)\}\)
B.\(F\{f(t)\} = sL\{g(t)\}\)
C.\(F\{f(t)\} = L\{g(t)\}\)
D.\(L\{f(t)\} = F\{g(t)\}\)

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: C

The true relation is F{f(t)} = L{g(t)}, with the transforms evaluated at matching arguments.

Taking the Fourier transform of \(f\) with kernel \(e^{-iyt}\):

\[ \begin{aligned} \mathcal{F}\{f\}(y) &= \int_{0}^{\infty} e^{-iyt}e^{-xt}g(t)\, dt \\ &= \int_{0}^{\infty} e^{-(x + iy)t}g(t)\, dt = \mathcal{L}\{g\}(x + iy) \end{aligned} \]

So the Fourier transform of \(f\) is the Laplace transform of \(g\) at \(s = x + iy\); this is how the Laplace transform is obtained from the Fourier transform. No factor \(e^{s}\) or \(s\) appears, which rules out A and B, and D has the two transforms the wrong way round.

Discuss in Forum

Loading…

Topics

Subject Categories

Tags: Laplace Transforms Questions and Answers || Laplace Transforms MCQ Questions and Answers || Laplace Transforms GK Questions and Answers || Laplace Transforms GK MCQ Questions || Laplace Transforms Multiple Choice Questions and Answers || Laplace Transforms GK || GK on Laplace Transforms || Engineering Mathematics Questions and Answers || Engineering Mathematics MCQ Questions and Answers || Engineering Mathematics GK Questions and Answers || GK on Engineering Mathematics

Laplace Transforms Questions and Answers

Topics

Subject Categories

About Us

Stay informed about all important Dates & Notifications