Separable Differential Equation

Example

Solve the separable first order equation: $\frac{dx}{dt} = x(1-x)$, with $x(0) = x_0$ and $0 < x_0 < 1$.

\[\begin{aligned} \frac{1}{x (1 - x)}dx &= dt \\ \int_{x_0}^{x} \frac{1}{x (1 - x)}dx &= \int_{0}^{t} dt \\ \text{Since } \frac{1}{x (1 - x)} &= \frac{1}{x} + \frac{1}{1 - x} \\ \int_{x_0}^{x} (\frac{1}{x} + \frac{1}{1 - x}) dx &= (ln(x) - ln(1 - x))\vert_{x_0}^{x} \\ (ln(x) - ln(1 - x)) - (ln(x_0) - ln(1 - x_0)) &= t \\ ln(\frac{x (1 - x_0)}{x_0 (1 - x)}) &= t \\ \frac{x (1 - x_0)}{x_0 (1 - x)} &= e^t \\ \frac{x}{1 - x} &= \frac{x_0 e^t}{1 - x_0} \\ \text{Add one to both sides:} \\ \frac{1}{1 - x} &= \frac{x_0 e^t + 1 - x_0}{1 - x_0} \\ 1 - x &= \frac{1 - x_0}{(1 - x_0) + x_0 e^t} \\ x &= 1 - \frac{1 - x_0}{(1 - x_0) + x_0 e^t} \end{aligned}\]