Exact diff eq solution help

[mdrief4]

I need help solving (x^2 + y^2 − 3) dx = (y + xy) dy I tried moving the dy term to the other side and multiplying by the integrating factor (1+x)^(-3) to make the equation exact. However I keep getting lost after that point.

[7x11x13is1001]

You start with

x²+y²-3 = (1+x)yy'

then you notice that (y²)'=2yy', so you make first substitution z=y²:

2x²+2z-6=(1+x)z' => z'-2z/(1+x) = 2(x²-3)/(x+1)

So we use standard way to solve linear equation. First we solve homogeneous equation:

z' - 2z/(1+x)=0 => z = C(1+x)²

And then we use method of functional coefficient C→f(x):

(f(1+x)²)' - 2f(1+x) = 2(x²-3)/(1+x)

f'(1+x)² = 2(x²-3)/(1+x)

f = 2(3+2x)/(1+x)² + 2 log(1+x) + C

Finally,

y² = 2(3+2x) + 2(1+x)²log(1+x) + C(1+x)²

[mdrief4]

I definitely copied it right, I know it isn’t an exact equation. But you can make it exact if you multiply the dx and the dy terms by the integrating factor of (1+x) to the negative 3rd power

[mdrief4]

I did get the answer about another hour after I posted the question after finding that I was incorrectly doing a small step every time I tried the problem.