Definition:Bernoulli's Equation

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

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

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

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

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

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

Substituting $$z = y^{1-n} = \frac 1 {y^{n-1}}$$ finishes it off.

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