Q4RP - Fundamentals Of Differential Equations And Boundary Value Problems

Question

In Problems 3–10, determine the Laplace transform of the given function.${\mathit{e}}^{\mathbf{3}\mathbf{t}}\mathit{s}\mathit{i}\mathit{n}\mathbf{4}\mathit{t}$

Step-by-Step Solution

Answer

$L\left\{e^{3t}sin4t\right\}\left(s\right)=\frac{4}{{(s-3)}^2+16}$

Step 1:Given Information

The given function is $e^{3t}sin4t$.

 

Step 2: Determining the Laplace transform:

Using the Laplace transform definition, $L\left\{e^{at}sinbt\right\}\left(s\right)=\frac{b}{{(s-a)}^2+b^2}$we get

$\begin{array}{c}L\left\{e^{3t}sin4t\right\}\left(s\right)=\frac{4}{{(s-3)}^2+4^2}\\ =\frac{4}{{(s-3)}^2+16}\end{array}$

Step 3: Determining the Result

Thus, the required Laplace transform is .$L\left\{e^{3t}sin4t\right\}\left(s\right)=\frac{4}{{(s-3)}^2+16}$