Let the population of rabbits surviving at a time t be governed by the differential equation\frac{dp \left(\right. t \left.\right)}{dt} \textrm{ } = \textrm{ } \frac{1}{2}p(t)

Engineering

Mathematics

Linear Differential Equation

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) equals

300 – 200

400 – 300

600 – 500

400 – 300

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$\frac{dp (t)}{dt} + (\begin{array}{l} - 1 \\ 2 \end{array}) pt = - 200$

$l . F . = e^{- \int \frac{1}{2} dt} = e^{\frac{- t}{2}}$

$P (t) e^{- t / 2} = \int (- 200) e^{\frac{- t}{2}} dt + C$

$= \frac{- 200 e^{- \frac{t}{2}}}{- \frac{1}{2}} + C$

P(t) e^–t/2 = 400 e^–t/2 + C

100 = 400 + C

C = – 300

P(t) = 400 – 300 e^t/2