Riya Andrews

2022-09-03

Using laplace trasnform find the convulation of (f*g)(t).
$let\phantom{\rule{thickmathspace}{0ex}}f\left(t\right)=\mathrm{sin}\left(3t\right)$ and $g\left(t\right)={e}^{-2t}$
Using laplace trasnform find the convulation of (f*g)(t).
convulation theorem: $h\left(t\right)=\left(f\ast g\right)\left(t\right)={\int }_{0}^{t}f\left(u\right)g\left(t-u\right)du$
$L\left(sin\left(3t\right)=\frac{3}{{s}^{2}+9}\phantom{\rule{0ex}{0ex}}L\left({e}^{-2t}\right)=\frac{1}{s+2}$
then how we processed for this problem

barquegese2

Let $h\left(t\right)=\left(f\ast g\right)\left(t\right)$. We have
$\mathcal{L}\left\{f\ast g\right\}=\mathcal{L}\left\{f\right\}\cdot \mathcal{L}\left\{g\right\}$
that is
$H\left(s\right)=F\left(s\right)G\left(s\right)=\frac{3}{{s}^{2}+9}\cdot \frac{1}{s+2}=-\frac{9}{13}\cdot \frac{s}{{s}^{2}+9}+\frac{6}{13}\cdot \frac{3}{{s}^{2}+9}+\frac{9}{13}\cdot \frac{1}{s+2}$
and then
$h\left(t\right)={\mathcal{L}}^{-1}\left\{F\left(s\right)G\left(s\right)\right\}=\left[-\frac{9}{13}\mathrm{cos}\left(3t\right)+\frac{6}{13}\mathrm{sin}\left(3t\right)+\frac{9}{13}{\mathrm{e}}^{-2t}\right]u\left(t\right)$

Do you have a similar question?