1. The problem statement, all variables and given/known data Find an equation that defines IMPLICITLY the parameterized family of solutions y(x) of the differential equation: 5xy dy/dx = x2 + y2 2. Relevant equations y=ux dy/dx = u+xdu/dx C as a constant of integration 3. The attempt at a solution I saw a similar D.E. solved using the y=ux substitution, but using it for mine wasn't as clean, so halfway through I decided to use an integrating factor: 5xy dy/dx = x2 + y2 dy/dx = [ x2 + y2 ] / 5xy u + x du/dx = [ x2 + u2x2 ] / 5x2u u + x du/dx = [ 1 + u2 ] / 5u Divide across by x (and switch the order of the LHS): du/dx + u/x = [ 1 + u2 ] / 5ux Integrating factor: (the integral symbol won't show up in the superscript) eINT[1/x]dx = eln(x) = x du/dx ⋅ x + u/x ⋅ x = [ 1 + u2 ] ⋅ x / 5ux d/dx [ux] = [ 1 + u2 ] / 5u dx Integrate both sides gets us: ux = x [ [ 1 + u2 ] / 5u ] + C After this, it's just cleaning up so the u's are on one side. We get: u - [ 1 + u2 ] / 5u = C/x [4u2 - 1] / 5u = C/x We can now resubstitute the u's for y/x: [4[y/x]2 - 1] / 5[y/x] = C/x 4y/5x - x/5y = C/x After this, I'm pretty stuck. I don't know how to isolate y(x) on one side. If I messed up anywhere, or if there is a much easier way to do this problem that I am ignoring, please let me know! Looking forward to the suggestions.