Ordinary Differential Equations/Bernoulli

An equation of the form

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

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

Its derivative is

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

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

The equation can be turned into

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

Or

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

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.