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Found in: Page 415

### 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

# In Problems 3–10, determine the Laplace transform of the given function.${{\mathbit{t}}}^{{\mathbf{2}}}{{\mathbit{e}}}^{\mathbf{-}\mathbf{9}\mathbf{t}}$

$L\left\{{t}^{2}{e}^{-9t}\right\}\left(s\right)=\frac{2}{{\left(s+9\right)}^{3}}$

See the step by step solution

## Step 1:Given Information.

The given function is .${t}^{2}{e}^{-9t}$

## Step 2: Determining the Laplace transform:

Using the Laplace transform definition, $L\left\{{t}^{n}{e}^{at}\right\}=\frac{n!}{{\left(s-a\right)}^{n+1}}$we get

$\begin{array}{c}L\left\{{t}^{2}{e}^{-9t}\right\}=\frac{2!}{{\left(s+9\right)}^{3}}\\ =\frac{2}{{\left(s+9\right)}^{3}}\end{array}$

## Step 3: Determining the Result

Thus, the required Laplace transform is .$L\left\{{t}^{2}{e}^{-9t}\right\}\left(s\right)=\frac{2}{{\left(s+9\right)}^{3}}$