Solution to Bernoulli's Equation

Theorem
Bernoulli's equation:
 * $(1): \quad \dfrac {\mathrm d y}{\mathrm d x} + P \left({x}\right) y = Q \left({x}\right) y^n$

where:
 * $n \ne 0, n \ne 1$

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

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

in $(1)$.

Then we have:

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

It has an integrating factor:

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.

Also see
When $n = 0$ or $n = 1$ the equation is linear, and Solution to Linear First Order Ordinary Differential Equation can be used.