Ordinary differential equations Definition

Separable equation

Builds on Ordinary differential equation Interval

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.

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Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Separable equation definition

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Tag	`0V4E623THY`
URI	`m2://0V4E623THY`
Kind	definition
Writing	W0 stub
MSC	34A05
Defines	Separable equation
Uses	Ordinary differential equation, Interval

Separable equation

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