Question 964357
{{{(x^2-y^2)dy=2xydx }}} is an example of an Euler-homogeneous DE of degree 2.

This can be solved by the substitution x = vy.
==> dx = vdy + ydv.

==> {{{(x^2-y^2)dy=2xydx }}} <==> {{{(v^2y^2-y^2)dy=2(vy)y(vdy+ydv) }}}

==> {{{(v^2 - 1)dy = 2v^2dy+2vydv}}}

<==> {{{-2vydv = (1+v^2)dy}}}

<==> {{{((2v)/(1+v^2))dv = -dy/y}}}

==> {{{ln(1+v^2) = -lny + lnC}}}
==> {{{ln(y(1+v^2)) = lnC}}}
==> {{{y(1+v^2) = C}}}
==> {{{y(1+x^2/y^2) = C}}}
==> {{{y+x^2/y = C}}}
==> {{{x^2 + y^2 = Cy}}}, the general solution to the original DE.