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

Answer 1

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

$\frac{dp}{dt}=\frac{p-400}{2}$
$\frac{dp}{p-400}=\frac{1}{2}dt$
In $|p-400|=\frac{1}{2}t+c$ at $t=0,p=100$
In $300=c$
In $|\frac{p-400}{300}|=\frac{t}{2}$
$\Rightarrow |p-400|=300e^{\frac{t}{2}}$
$\Rightarrow 400-p=300e^{\frac{t}{2}}$ (as $p<400$)
$\Rightarrow p=400-300e^{\frac{t}{2}}$