Ordinary Differential Equations/Bernoulli

< Ordinary Differential Equations

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)


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

Which is linear.

Jacobi EquationEdit

The Jacobi equation


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