Seperable Equations using a substitution for a differential equation

[laura_a]

Homework Statement

The d.e

y' = (y+2x)/(y-2x)

is NOT seperable, but if you use a substitution then you obtain a new d.e involving x and u, then the new d.e is seperable... Solve the original d.e by using this change of variable method

Homework Equations

I'm going to use the substitution that u=y/x in the form y=ux

The Attempt at a Solution

y' = (y+2x)/(y-2x)
let y=ux
then

y' = (ux+2x)/(ux-2x)
y'(ux-2x) - (ux+2x) = 0 <--- thus it is seperable
So I can say

d/dx((ux^2)/2 - x^2) - d/dx((ux^2/2) + x^2) = 0

Putting it all together

d/dx[(ux^2)/2 - x^2 - (ux^2)/2 - x^2] = 0
d/dx (-2x^2) = 0

Well that is where I am stuck, how do I solve it from there and what am I trying to get because I've changed the d.e so I'm not sure how the answer from the new d.e will help me find a solution to the original one?

 

[Dick]

If you are going to substitute y=ux then you also need to substitute y'=(ux)'=u'x+u. That will change the differential equation for y into one for u. You shouldn't have a y' hanging around after the substitution.