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Let the population of rabbits surviving ...

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

A

(a) `400-300e^(t/2)`

B

(b) `300-200e^(t/2)`

C

(c) `600-500e^(t/2)`

D

(d) `400-300e^(t/2)`

Text Solution

Verified by Experts

The correct Answer is:
(a)
Given, differential equation is `(dp)/dt-1/2p(t)=-200` a
linear differential equation.
Here, `p(t)=(-1)/2,Q(t)=-200`
`IF=e^(int-(1/2)dt)=e^(t/2)`
Hence, solution is
`p(t) cdot IF = int Q(t) cdot IF dt`
`p(t) cdot e^(t/2)=int-200 cdot e ^(t/2)dt`
`p(t) cdot e^(t/2)=400 cdot e ^(t/2)+k`
`rArr p(t)=400 +ke^(-1//2)`
If p(0)=100, then k=-300
`rArr p(t)=400-300e^(1/2)`
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