Definition:Bernoulli's Equation

Theorem
Bernoulli's differential equation is a first order ordinary differential equation which can be put into the form:
 * $\displaystyle \frac {\mathrm d y}{\mathrm d x} + P \left({x}\right) y = Q \left({x}\right) y^n$

where $n \ne 0$ and $n \ne 1$.

It has the general solution:
 * $\displaystyle \frac {\mu \left({x}\right)} {y^{n-1}} = \left({1-n}\right) \int Q \left({x}\right) \mu \left({x}\right) \ \mathrm d x + C$

where:
 * $\mu \left({x}\right) = e^{\int \left({1-n}\right) P \left({x}\right) \ \mathrm d x}$

Proof
Make the substitution:
 * $z = y^{1-n}$

in the original equation.

Then we have:

This is now a linear first order ordinary differential equation in $z$.

It has an integrating factor:
 * $\mu \left({x}\right) = e^{\int \left({1-n}\right) P \left({x}\right) \ \mathrm d x}$

and this can be used to obtain:
 * $\displaystyle \mu \left({x}\right) z = \left({1-n}\right) \int Q \left({x}\right) \mu \left({x}\right) \ \mathrm d x + C$

Substituting $z = y^{1-n} = \dfrac 1 {y^{n-1}}$ finishes the proof.

Note
When $n = 0$ or $n = 1$ the equation is already linear, and the technique for solving that can be used.