Originally Posted by

**MarkFL** Continuing where I left off...

$\displaystyle \mu(t)=\exp\left(-r\int\,dt\right)=e^{-rt}$

And thus the ODE becomes:

$\displaystyle \frac{d}{dt}\left(e^{-rt}A\right)=Pe^{-rt}$

Integrate:

$\displaystyle e^{-rt}A=-\frac{P}{r}e^{-rt}+c_1$

$\displaystyle A(t)=-\frac{P}{r}+c_1e^{rt}$

Now, we are given:

$\displaystyle A(0)=-\frac{P}{r}+c_1=0\implies c_1=\frac{P}{r}$

Hence, the solution to the IVP is:

$\displaystyle A(t)=-\frac{P}{r}+\frac{P}{r}e^{rt}=\frac{P}{r}\left(e^{ rt}-1\right)$

Can you proceed?