Answer to Question #262972 in Differential Equations for Neilmar

Question

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

dy/dx = y(xy^3-1)

Answer 1

Solution

Let’s rewrite equation in the form

dy/dx + y = xy^4

New function u(x) = y^-3   

du/dx = -3 y^-4 (dy/dx)   => dy/dx = - (du/dx) y⁴ /3

Substitution in equation gives new equation for u:

du/dx – 3u = –3x

It is linear equation . So  u(x) = C e^3x + Ax + B, where A, B, C – are arbitrary constants,  C e^3x – is general solution of homogeneous  equation.

Substitution into equation on u gives:

A – 3B – 3Ax = -3x => A = 1, B = 1/3.

Therefore u(x) = C e^3x + x + 1/3 and

$y(x)=\sqrt[3]{\frac{1}{u(x)}}=\frac{1}{\sqrt[3]{Ce^{3x}+x+1/3}}$