Chapter 7.4, Problem 6P

Program Plan Intro

Program Description: Purpose of problem is to obtain Laplace transform of given function $f\left(t\right)=e^{at},g\left(t\right)=e^{bt}\left(a\ne b\right)$ .

Summary introduction:Program will use the convolution theorem in which $f\left(t\right)$ and $g\left(t\right)$ are piecewise continuous for $t\ge 0$ . Here, $\left|f\left(t\right)\right|$ and $\left|g\left(t\right)\right|$ are bounded by $M{e}^{ct}$ as $t\to \infty$ .The Laplace transform of the convolution $f\left(t\right)\ast g\left(t\right)$ is expressed as $L\left\{f\left(t\right)\ast g\left(t\right)\right\}=L\left\{f\left(t\right)\right\}\cdot L\left\{g\left(t\right)\right\}$ or $L^{-1}\left\{F\left(s\right)\cdot G\left(s\right)\right\}=f\left(t\right)\ast g\left(t\right)$ .

The inverse of the function $L^{-1}\left\{F\left(s\right)\cdot G\left(s\right)\right\}$ is defined as,

  $f\left(t\right)\ast g\left(t\right)={\displaystyle \underset{0}{\overset{t}{\int }}f\left(\tau \right)}g\left(t-\tau \right)d\tau$ ....... (1)

Here, the Laplace transform of the convolution $f\left(t\right)\ast g\left(t\right)$ exist for $s>c$ .