Ordinary differential equations Definition

Linear first-order equation

Builds on Ordinary differential equation Interval

Definition

Let $I$ be an open interval and let $p, q : I \to \mathbb{R}$ be continuous functions. The linear first-order equation with coefficient $p$ and right side $q$ is the equation $y' + p(x)\, y = q(x)$ for an unknown differentiable function $y : I \to \mathbb{R}$ . It is homogeneous if $q = 0$ on $I$ , and inhomogeneous otherwise.

Remarks

The solution formula gives each solution explicitly, with an integrating factor. The homogeneous equation $y' + p(x) y = 0$ has the solutions $y = C e^{-P(x)}$ , where $P$ is an antiderivative of $p$ and $C$ is a constant. Many authors write the equation as $y' = a(x) y + b(x)$ , with $a = -p$ and $b = q$ .

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Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Linear first-order equation definition

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Tag	`0Z3YKRQZ48`
URI	`m2://0Z3YKRQZ48`
Kind	definition
Writing	W0 stub
MSC	34A30
Defines	Linear first-order equation
Uses	Ordinary differential equation, Interval

Linear first-order equation

Definition

Remarks

Reading path

Where this leads

Further reading

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