Solve for $N(t)$ (seperation of variables)

Question

Related to Bernoulli smallpox model

$\frac{dN(t)}{dt}=-\mu N(t)-\gamma N_0 e^{-(\mu +1)t}$ with initial condition $N_0=N(0)$

I rearrange it as

$\frac{dN(t)}{dt}+\mu N(t)=-\gamma N_0 e^{-(\mu +1)t}$ then what...

Answer 1

It checks for me: \begin{align*} N(t) &= N_0\frac{e^{-mt}(m+r)-r}{m}, \\ \dot N(t) &= -N_0e^{-mt}(m+r); \\ -mN(t) &= -N_0e^{-mt}(m+r)+rN_0, \\ -mN(t)-rN_0 &= -N_0e^{-mt}(m+r). \end{align*}