Bernoulli 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 $n = 2$.

Explanations

L1 human

A Bernoulli equation is a linear equation with an extra power of $y$ on the right side. It is not linear, but a change of the unknown makes it linear. Then the solution formula for linear equations solves it.