# What is the logistic equation that satisfies the initial condition of (0,4) where the logistic differential equation is dy/dt=y(1-y/36)?

Jan 21, 2018

$y = \frac{36}{1 + 8 {e}^{- t}}$

#### Explanation:

The general form of the differential equation is:

$\frac{\mathrm{dy}}{\mathrm{dt}} = k \cdot y \left(1 - \frac{y}{L}\right)$

in that form the solution is of the form:

$y = \frac{L}{1 + A {e}^{- k t}}$ where $A = \frac{L - {y}_{0}}{y} _ 0$.

(Arrived at through partial fraction decomposition and separation of variables. See this video for the solution. )

For this problem $k = 1$, $L = 36$, and ${y}_{0} = 4$.

Calculating $A$:

$A = \frac{36 - 4}{4} = 8$.

Substituting everything:

$y = \frac{36}{1 + 8 {e}^{- t}}$