Question

solve:
(x-y)y'=2x+y

Answer 1

I think this is partial differential equation. I'm still taking the class so I'll see what I can do.

I would first rewrite it as \[\frac{ dy }{ dx } (x-y) = 2x + y\]

Then you can separate the variables and times both sides with dx to get:
(x-y) dy = 2x + y
(-y - y) dy = (2x - x) dx
-2y dy = x dx

Then you can integrate both sides and solve for y. Hope this helps.

Answer 2

I'm sorry the first line to my working should have been (x-y) dy = (2x+y) dx

Answer 3

Try substituting \(y=xv\), so that \(y'=v+xv'\):
\[(x-y)y'=2x+y~~\iff~~(x-xv)(xv'+v)=2x+xv\]
Simplifying a bit, you get something like
\[v'=\frac{v^2+2}{x(1-v)}\]
So a fairly simple separable equation.