posted by anonymous .
solve the following differential equation: (x^2+xy)dy=(xy-y^2)dx
See if this works:
It does not look like it meets the "exact differential" requirements
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 = Cex/y
Check my math, as always.
I want to congratulate and thank Steve for that very impressive solution.