Ordinary Differential Equations/Bernoulli

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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[edit]

The Jacobi equation

(a_1+b_1x+c_1y)(xdy-ydx)-(a_2+b_2x+c_2y)dy+(a_3+b_3x+c_3y)dx=0

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