Solution to Bernoulli's Equation

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

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

has the general solution:
 * $\displaystyle \frac {\map \mu x} {y^{n - 1} } = \paren {1 - n} \int \map Q x \map \mu x \rd x + C$

where:
 * $\map \mu x = e^{\paren {1 - n} \int \map P x \rd x}$

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:
 * $\displaystyle \map \mu x z = \paren {1 - n} \int \map Q x \map \mu x \rd x + C$

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.