4. Separable Differential Equations

Separable Differential Equations

Definition

A (first-order) separable differential equation is a first-order differential equation in which the expression for $y^{\prime }=\frac{dx}{dy}$ can be factored as a function of $x$ times a function of $y$.

In other words, it can be written in the form

$\frac{dy}{dx}=f(x)g(y)=\frac{f(x)}{h(y)}$

Differential Form

$h(y)dy=f(x)dx$

Solution Procedure - Algorithm

Algorithm to solve separable differential equations:

1. write DE as: $\frac{dy}{dx}=\frac{f(x)}{h(y)}$
2. Separate variables: $h(y)dy=f(x)dx$
3. Find anti-derivatives: $H(y)=\int h(y)dy=\int f(x)dx=F(x)+C$
4. Implicit solution: $H(y)=F(x)+C$ with $C\in R$
5. Explicit solution: $y(x)=H^{-1}(F(x)+C)$ with $C\in R$