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

Question

Let the population of rabbits surviving at a time t be governed by the differential equation (dp(t)/dt)= (1/2) p(t)-200. If p(0)=100, then p(t) is equal to (A) 400-300 et/2 (B) 300-200 e-t/2 (C) 600-500 et/2 (D) 400-300 et/2

Tardigrade Admin · Accepted Answer

(d p(t)/d t)=(1/2) p(t)-200 ∫ (d(p(t))/((1)2 p(t)-200))=∫ dt 2 log ((p(t)/2)-200)=t+c (p(t)/2)-200=e(t/2) k Using given condition p(t)=400-300 et / 2

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

$400 - 300 e^{t /2}$

$300 - 200 e^{- t /2}$

$600 - 500 e^{t} /2$

$400 - 300 e^{t} /2$

$\frac{d p ( t )}{d t} = \frac{1}{2} p (t) - 200$
$\int \frac{d ( p ( t ))}{( \frac{1}{2} p ( t ) - 200 )} = \int d t$
$2 lo g (\frac{p ( t )}{2} - 200) = t + c$
$\frac{p ( t )}{2} - 200 = e^{\frac{t}{2}} k$
Using given condition $p (t) = 400 - 300 e^{t /2}$

Q. Let the population of rabbits surviving at a time t be governed by the differential equation dtdp(t)​=21​p(t)−200. If p(0)=100, then p(t) is equal to

400−300et/2

300−200e−t/2

600−500et/2