# Question #19983

Jun 23, 2017

$P = - \frac{C {e}^{k M t}}{1 - C {e}^{k M t}}$

#### Explanation:

Assuming the $M$ is a constant, you would need to first separate the variables so that

$\frac{1}{k P \left(M - P\right)} \mathrm{dP} = \mathrm{dt}$

Integrate both sides of this equation

$\int \left[\frac{1}{k P \left(M - P\right)}\right] \mathrm{dP} = \int \mathrm{dt}$

$\frac{\ln \left(P\right) - \ln \left(P - M\right)}{k M} = t + C$

$\ln \left(\frac{P}{P - M}\right) = k M t + C$

Exponentiate both sides with $e$

${e}^{\ln \left(\frac{P}{P - M}\right)} = {e}^{k M t + C}$

$\frac{P}{P - M} = C {e}^{k M t}$

$P = C {e}^{k M t} \left(P - M\right)$

$P = P C {e}^{k M t} - C {e}^{k M t}$

$P - P C {e}^{k M t} = - C {e}^{k M t}$

$P \left(1 - C {e}^{k M t}\right) = - C {e}^{k M t}$

$P = - \frac{C {e}^{k M t}}{1 - C {e}^{k M t}}$

Jun 23, 2017