Question

Let the population of rabbits surviving at a time $t$ be governed by the differential equation
$\frac {dp(t)}{dt}=\frac {1}{2} p(t)-200.$ If $p(0)=100,$ then $p(t)$ is equal to

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

Answer 1

Answer: (A)

$\frac{d p(t)}{d t}=\frac{1}{2} p(t)-200$
$\int \frac{d(p(t))}{\left(\frac{1}{2} p(t)-200\right)}=\int dt$
$2 \,\log \left(\frac{p(t)}{2}-200\right)=t+c$
$\frac{p(t)}{2}-200=e^{\frac{t}{2}} k$
Using given condition $p(t)=400-300 \,e^{t / 2}$