Question

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:

• A. $400-300e^{t/2}$
• B. $300-200e^{-t/2}$
• C. $600-500e^{t/2}$
• D. $400-300e^{-t/2}$

Answer 1

The correct option is A $400-300e^{t/2}$

$\frac{dp\left(t\right)}{dt}=\frac{1}{2}p\left(t\right)-200$
$\Rightarrow 2\frac{dp\left(t\right)}{dt}=p-400$
$\Rightarrow 2\frac{dp\left(t\right)}{p-400}=dt$
$\Rightarrow 2\cdot \ln (p-400)=t+c$
$\Rightarrow \ln (p-400)=\frac{t}{2}+c...\left(1\right)$
At $t=0,p=100$
$\Rightarrow \ln |100-400|=c$
$\Rightarrow c=\ln |300|$
put value of $c$ in equation (1)
$\Rightarrow \ln \left|\frac{p-400}{300}\right|=\frac{t}{2}$
$\Rightarrow p=400-300e^{t/2}$