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Q13E

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Found in: Page 404

### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

# Find the Laplace transform of ${\mathbit{f}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}{\mathbf{=}}\underset{0}{\overset{t}{\int }}\left(t-v\right){e}^{3v}dv$

Laplace transformation of${\int }_{0}^{t}{e}^{v}\mathrm{sin}\left(t-v\right)dv$ is $\frac{1}{{s}^{2}\left(s-3\right)}$.

See the step by step solution

## Step 1: Assuming g(t) and h(t).

Let

$\begin{array}{l}g\left(t\right)=t\\ h\left(t\right)={e}^{3t}\end{array}$

## Step 2: Getting the values of G(s) and H(s).

Now,

$\begin{array}{c}G\left(s\right)=\mathcal{L}\left\{g\left(t\right)\right\}\left(s\right)\\ =\frac{1}{{s}^{2}}\end{array}$

$\begin{array}{c}H\left(s\right)=\mathcal{L}\left\{h\left(t\right)\right\}\left(s\right)\\ =\frac{1}{s-3}\end{array}$

## Step 3: Now, On applying the Convolution Theorem

We get,

$\begin{array}{c}\mathcal{L}\left\{f\left(t\right)\right\}\left(s\right)=\mathcal{L}\left\{{\int }_{0}^{t}\left(t-v\right){e}^{3v}dv\right\}\left(s\right)\\ =\mathcal{L}\left\{{\int }_{0}^{t}g\left(t-v\right)h\left(v\right)dv\right\}\end{array}$

$\begin{array}{c}\mathcal{L}\left\{f\left(t\right)\right\}\left(s\right)=\mathcal{L}\left\{g\left(t\right)\ast h\left(t\right)\right\}\left(s\right)\\ =G\left(s\right)\cdot H\left(s\right)\\ =\frac{1}{{s}^{2}\left(s-3\right)}\end{array}$

Therefore,

$\mathcal{L}\left\{f\left(t\right)\right\}\left(s\right)=\frac{1}{{s}^{2}\left(s-3\right)}$