Differential Equations/Bernoulli

Current revision (unreviewed)

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.

The Jacobi equation

$(a 1 + b 1 x + c 1 y)(x d y - y d x) - (a 2 + b 2 x + c 2 y) d y + (a 3 + b 3 x + c 3 y) d x = 0$

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