Let the population of rabbits surviving' at a time $t$ be governed by the differential equation $\frac{dp\left(t\right)}{dt}=\frac{1}{2}p\left(t\right)-200.$ If $p\left(0\right)=100$ then $p\left(t\right)$ equals

$\frac{dp}{dt}=\frac{1}{2}p\left(t\right)-200\Rightarrow \frac{dp}{p-400}=\frac{1}{2}dt$

Integrating, we get, $\ln |p-400|=\frac{1}{2}t+c$

$t=0,p=100\Rightarrow \ln 300=c$

Again $\ln \left(\frac{p-400}{300}\right)=\frac{t}{2}\Rightarrow |p-400|=300e^{t/2}$

$\begin{array}{l}\therefore 400-p=300e^{t/2}(p<400)\\ \therefore p=400-300e^{t/2}\end{array}$