## Bernoulli Equations

Bernoulli equations are of the form:

Dividing by $y^n$:

If $n=0$ or $n=1$ then the equation is linear.  Otherwise a substitution of $u=y^{1-n}$ will give us a linear equation.

If $u=y^{1-n}$, then $\frac{du}{dx}u=(1-n)\frac{dy}{dx}y^{-n}$.  Substituting these into the second equation from above:

For example,

Dividing by $y^4$, we get  $y'y^{-4}+2xy^{-3}=x$

Substituting $u=y^{-3}$ and $\frac{du}{dx}={-3}\frac{dy}{dx}y^{-4}$ gives us