Question

Find theLaplace transform $\mathrm{f}\times \mathrm{g}$of the given functions. Do not evaluate convolutionintegral before transforming where $\mathbf{L}\left\{e^{2t}*\mathrm{sint}\right\}$

Short answer

The laplace transformation is$L\left\{e^{2t}*sint\right\}=\left[\frac{1}{\left(s-2\right)(s^2+1)}\right]$

Step-by-step solution

Laplace transformation of the given function;

$\begin{array}{l}L\left\{e^{2t}*\sin t\right\}\\ L\left\{e^{2t}*e^t\sin t\right\}=L\left\{e^{2t}\right\}.L\left\{\sin t\right\}\end{array}$

Using the formula to solve further;

If function f and g are piecewise continuous on the interval then

$\begin{array}{l}L\left\{e^{at}f\left(t\right)\right\}=F{(s\left.)\right|}_{s\to s-a}\\ ={\left.\frac{1}{s}\right|}_{s\to s-a}.\frac{1}{s^2+1}\\ =\frac{1}{s-2}.\frac{1}{s^2+1}\\ =\frac{1}{\left(s-2\right)\left(s^2+1\right)}\end{array}$

Hence the final answer is$L\left\{e^{2t}*sint\right\}=\left[\frac{1}{\left(s-2\right)(s^2+1)}\right]$