Answer to Question #147611 in Differential Equations for fanni

Question

The rate at which a population of lions at etosha national park P(t) is progressing is given by the differential equation
dp/dt=P(M-kP) where M, k are positive constants
Solve the differential equation to determine an expression for P(t)

Answer 1

$-\int \frac{dp}{p(kp-M)}=\int dt,$

$\frac 1 M ln|\frac{kp-M}{p}|=t+\frac 1MlnC_1,$

$ln|\frac{kp-M}{C_1p}|=Mt,$

$\frac{k}{C_1}-e^{Mt}=\frac{M}{C_1p},$

$p(t)=\frac{M}{k-C_1e^{Mt}}=\frac{M}{k-e^{Mt+C}}.$