Solve Non-Linear ODE with a Substitution | Step-by-Step Guide

[SeReNiTy]

Hi guys, just wondering if you can give a hand on a non-linear ODE, all you guys need to do if give me a good substituion to try...

xy''+y'+(y')^3 = 0

where ' = d/dx

 

[HallsofIvy]

Do you have any reason to believe there is a "good substitution"?

Almost all non-linear differential equations cannot be solved exactly.

 

[SeReNiTy]

Because this is a problem in a text, so i believe when thye say find the exact analytical solution, they must mean one exists...

 

[HallsofIvy]

Okay, that's a good reason! Also I just noticed that you have y' twice and no y. If you let z= y', you can reduce the order of the equation:
xy''+y'+(y')^3 = 0 becomes xz'+ z+ z³= 0, a first order, separable equation. Solve for z and let y'= z.