Chapter 10.10, Problem 32AP

Let $y\left(t\right)$ be the solution to the initial-value problem $y^{\prime }+ay=f\left(t\right)$, $y\left(0\right)=y_0$, where $a$ and $y_0$ are constants. Verify that

$L\left[y\right]=\frac{L\left[f\right]}{s+a}+\frac{y_0}{s+a}$

And show that

$y\left(t\right)=y_0{e}^{-at}+{\displaystyle {\int }_0^t{e}^{-a\left(t-w\right)}f\left(w\right)dw}$.