Discussion Forum

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 

Answer:

Explanation:

Given, differential  equation $\frac{dp}{dt}-\frac{1}{2}p(t)=-200$ is linear ndifferential equation 

 Here,   $p(t)=-\frac{1}{2}, Q(t)= -200$

$IF=e^{\int_{}^{} -(\frac{1}{2})dt}=e^{-t/2}$

 Hence, solution is

          $p(t).IF=\int_{}^{} Q(t)IF dt$

   $p(t).e^{-t/2}=\int_{}^{} -200.e^{-\frac{t}{2}dt}$

                $p(t).e^{-t/2}= 400.e^{-\frac{t}{2}}+K$

           $\Rightarrow$   $ p(t).= 400+ke^{-\frac{1}{2}}$

if p(0)=100, then k=-300

$\Rightarrow$   $400-300e^{\frac{t}{2}}$