The Bernoulli Differential Equation
A classical nonlinear equation tamed by a single substitution
1. What Is a Bernoulli Equation?
Standard Form
A first-order ODE is called a Bernoulli equation if it can be written as:
where $P(x)$ and $Q(x)$ are continuous functions and $n$ is a real number. The equation is named after Jacob Bernoulli, who first discussed it in 1695, though the solution method we use today was published by Gottfried Leibniz in the same year.
Notice the structure: the left side looks exactly like a first-order linear ODE, but the right side carries a power $y^n$ that breaks linearity. That single power term is what makes the equation nonlinear — and also what makes it solvable by a clever trick.
Special Exponent Values — Dispatch Immediately
| Exponent $n$ | What happens | Method |
|---|---|---|
| $n = 0$ | $y^0 = 1$, so the equation becomes $\frac{dy}{dx} + P(x)y = Q(x)$ — a standard first-order linear ODE. | Integrating factor (already learned). |
| $n = 1$ | $y^1 = y$, so the equation becomes $\frac{dy}{dx} + P(x)y = Q(x)y$, i.e. $\frac{dy}{dx} = [Q(x) - P(x)]y$ — a separable equation. | Separate and integrate. |
| $n \neq 0, 1$ | The equation is genuinely nonlinear. This is the new territory. | Bernoulli substitution (this lesson). |
Recognition Gallery — Spotting Disguised Bernoulli Equations
Bernoulli equations don't always arrive in standard form. You may need to divide or rearrange first.
| Given Form | Divide / Rearrange | Standard Form | $n$ |
|---|---|---|---|
| $x\frac{dy}{dx} + y = y^2 x^2 \ln x$ | Divide by $x$ | $\frac{dy}{dx} + \frac{1}{x}y = (x\ln x)\,y^2$ | $2$ |
| $2\frac{dy}{dx} + (\tan x)\,y = \frac{(4x+5)^2}{\cos x}\,y^3$ | Divide by $2$ | $\frac{dy}{dx} + \frac{\tan x}{2}\,y = \frac{(4x+5)^2}{2\cos x}\,y^3$ | $3$ |
| $y' + \frac{y}{x} - \sqrt{y} = 0$ | Move $\sqrt{y}$ to right | $\frac{dy}{dx} + \frac{1}{x}y = y^{1/2}$ | $\frac{1}{2}$ |
| $y' - y = e^{2x}y^3$ | Already in form | $\frac{dy}{dx} - y = e^{2x}\,y^3$ | $3$ |
| $y' + 3y = xy^{-2}$ | Already in form | $\frac{dy}{dx} + 3y = x\,y^{-2}$ | $-2$ |
Key craft: Read off $n$ only after the equation is in standard form $y' + P(x)y = Q(x)y^n$.