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Q.
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) equals
Differential Equations
Solution:
Since $\frac{dp}{dt} - \frac{1}{2} p\left(t\right) = -200$ is lmear in $y$
$\therefore I.F. = e^{\int \frac{1}{2} dt} = e^{-\frac{t}{2}}$
$\therefore $ role is $p\cdot e^{\frac{-t}{2}} = \int -200\cdot\left(e^{-\frac{t}{2}}\right)dt + C$
$= -200\cdot \frac{e^{-t/ 2}}{-1/ 2}+C$
$= 400 \,e^{-t/2} + C$
Since $p\left(0\right) = 100$
$\therefore 100 \,e^{0} = 400\, e^{0} + C$
$\Rightarrow 100 = 400 + C$
$\Rightarrow C = - 300$
$\therefore p\left(t\right) = 400 - 300 \,e^{t/2}$