Ordinary differential equations Definition

Bernoulli equation

Builds on Ordinary differential equation Linear first-order equation

Definition

Let $I$ be an open interval, let $P, Q : I \to \mathbb{R}$ be continuous functions, and let $n$ be a real number. A Bernoulli equation is an ordinary differential equation of the form $y' + P(x)\, y = Q(x)\, y^{n}$ for an unknown function $y$ on a subinterval of $I$ . If $n$ is not an integer, the function must satisfy $y > 0$ .

Remarks

For $n = 0$ and $n = 1$ the equation is linear. Some authors exclude these two values. For $n \ne 1$ , the substitution $v = y^{1 - n}$ changes the equation, on the set where $y \ne 0$ , into the linear equation $v' + (1 - n) P(x)\, v = (1 - n) Q(x)$ . For $n > 0$ , the function $y = 0$ is also a solution. The logistic equation is a Bernoulli equation with $n = 2$ .

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

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Tag	`0ZB84QCDKR`
URI	`m2://0ZB84QCDKR`
Kind	definition
Writing	W1 stated
MSC	34A34
Defines	Bernoulli equation
Uses	Ordinary differential equation, Linear first-order equation

Bernoulli equation

Definition

Remarks

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