Differential Eq using Substitution help

[g.sharm89]

Hey. I am having a hard time solving this problem.


(x^2)y' + 2xy = 5y^4

I get as far as simplifying to
y' = [(5y^4)/(x^2)] - 2y/x

Then use v: y/x and y: vx & y': v'x + v

And get

v'x + v = [5(v^4)(x^2)] - 2v


And then I get lost. Any help would be appreciated. Thanks!

[JJacquelin]

The convenient substitution is v=1/y^3

[LawlQuals]

Agreed. while v = y/x does work quite frequently (at least in your homework problems I imagine), you have probably learned about other trademark substitutions of different types of equations. Specifically, this is a Bernoulli type equation (and, it has its own characteristic substitution, and JJacquelin has informed which specific substitution to use in this problem). Learning to diagnose a differential equation and determine what type it is, is a quick way to know immediately how to solve them. I encourage you to look up Bernoulli differential equations online, you will find a lot of material.