Separable equation

Definition

Let $I$ and $J$ be open intervals of $\mathbb{R}$, and let $g : I \to \mathbb{R}$ and $h : J \to \mathbb{R}$ be continuous functions. A separable equation is a first-order ordinary differential equation of the form $$y' = g(x)\, h(y),$$ for an unknown function $y$ of $x$ with values in $J$.

Remarks

Method: if $h(y) \ne 0$, write the equation as $y'/h(y) = g(x)$ and integrate. With antiderivatives $H$ of $1/h$ and $G$ of $g$, the solutions satisfy $H(y(x)) = G(x) + C$ for a constant $C$. For each zero $c$ of $h$, the constant function $y = c$ is a solution. The method above does not find these solutions. Example: $y' = x y$ has the solutions $y = C e^{x^2/2}$ with a real constant $C$. Exponential growth and the logistic equation are separable.

Explanations

L1 human

In a separable equation, the right side is a product: one factor depends only on $x$ and the other only on $y$. Put everything with $y$ on one side and everything with $x$ on the other side, then integrate both sides.