Ordinary Differential Equations/Bernoulli

An equation of the form

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

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

Its derivative is

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

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

The equation can be turned into

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

Or

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

Which is linear.

Jacobi Equation

The Jacobi equation

$(a_1+b_{1}x+c_{1}y)(xdy-ydx)-(a_2+b_{2}x+c_{2}y)dy+(a_3+b_{3}x+c_{3}y)dx=0$

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