Bernoulli differential equation

Definition

In normalized form, this first-order first-degree differential equation looks like:

$y^{\prime }+p(x)y=q(x)y^n$

where $n\ne 0,1$. (Note that the cases $n=0,1$ give first-order linear differential equations).

Solution method and formula

Divide both sides by $y^n$. If $n>0$, this means that we may be potentially discarding the stationary solution $y=0$, and must remember to add that back to the solution family at the end.

We get:

$\frac{y^{\prime }}{y^n}+\frac{p(x)}{y^{n-1}}=q(x)$

Now put $w=1/y^{n-1}$ to get:

$\frac{w^{\prime }}{1-n}+p(x)w=q(x)$

Multiply by $1-n$ to get:

$w^{\prime }+(1-n)p(x)w=(1-n)q(x)$

This is now a first-order linear differential equation in $w$, and can be solved to get a family of functional solutions for $w$ in terms of $x$. Plugging back $w=1/y^{n-1}$ gives a family of functional solutions for $y$ in terms of $x$. We can now add back $y=0$.