Solve a Bernoulli Equation

Question

Solve the given differential equation by using an appropriate substitution. The DE is a Bernoulli equation.

dy/dx = y(xy2 − 1)

Answer 1

dy/dx + y = xy³. Substitution v = y^-2; dv/dx = - 2y^-3dy/dx; dy/dx = - 1/2y³dv/dx; - 1/2y³dv/dx + y = xy³_;

-1/2dv/dx + 1/y² = x; dv/dx - 2v = - 2x; M = e^∫-2dx = e^-2x; (e^-2x·v)' = - 2x; e^-2x·v = - x² + C; v = (- x² + C)e^2x;

y² = e^-2x/(C - x²)