86 Example

To solve the integral equation

$$y(t)=2e^{at}+{\int }_0^t{e}^{b(t-u)}y(u)𝑑u$$

where $a,b$ are constants with $a\ne b+1$, and $y$ satisfies $(E)$. Solution. Note that the the integral is a convolution of $y$ with $e^{bt}$, so by Proposition 73(iii), we have

$$\widehat{y}(s)=\frac{2}{s-a}+\frac{\widehat{y}(s)}{s-b};$$

after a little reduction we obtain

$$\widehat{y}(s)=\frac{2(s-b)}{(s-a)(s-b-1)};$$