Question

solve the following differential equation: (x^2+xy)dy=(xy-y^2)dx

Answer 1

See if this works:
http://www.math.colostate.edu/~gerhard/M345/CHP/ch2_6.pdf

It does not look like it meets the "exact differential" requirements

Answer 2

Good reference. Try looking at the last example:

Substitute y = xv ⇒ dy = v dx+x dv

(xy-y^2)dx - (x^2+xy)dy = 0
(x^2v - x^2v^2)dx - (x^2 + x^2v)(vdx + xdv) = 0

2/x dx + (1+v)/v^2 dv = 0

2lnx + (lnv - 1/v) = C

ln(x^2v) - 1.v = C

vx^2 = Cexp(1/v)
y = vx

xy = Ce^x/y

Check my math, as always.

Answer 3

I want to congratulate and thank Steve for that very impressive solution.