# Ordinary Differential Equations/Bernoulli

An equation of the form

${\displaystyle {dy \over dx}+f(x)y=g(x)y^{n}}$

can be made linear by the substitution ${\displaystyle z=y^{1-n}}$

Its derivative is

${\displaystyle {dz \over dx}=(1-n)y^{-n}{dy \over dx}}$

So that multiplying it by ${\displaystyle y^{-n}}$

The equation can be turned into

${\displaystyle {dy \over dx}y^{-n}+f(x)y^{1-n}=g(x)}$

Or

${\displaystyle {dz \over dx}+(1-n)f(x)z=(1-n)g(x)}$

Which is linear.

### Jacobi EquationEdit

The Jacobi equation

${\displaystyle (a_{1}+b_{1}x+c_{1}y)(xdy-ydx)-(a_{2}+b_{2}x+c_{2}y)dy+(a_{3}+b_{3}x+c_{3}y)dx=0}$

can be turned into the Bernoulli equation with the appropriate substitutions.