I'm not finding any resource or description or systemic methodology to find integrating factors when the integrating factor will be a function of both x and y.
I'm on this problem,
$$ ( y - xy^2 ) dx + (x + x^2y^2) dy = 0 $$
In its present form its not exact, but multiplying through by an integrating factor $ \frac{1}{x^2y^2} $ does make it exact.
Now I want to know how you find that, when the integrating factor is both a function of x and y.
The technique on this page for finding an integrating factor $u(x)$ fails.
Which means there's no function of purely x that will make this equation exact.
So I'm trying something like this:
1. Multiply the original equation by a function u(x,y):$$ u(x,y)( y - xy^2 )dx + u(x,y)(x + x^2y^2)dy = 0 $$
2. Attempt to set $ \frac{\partial M(x,y)}{\partial y} = \frac{\partial N(x,y)}{\partial x} $$$ \frac{\partial}{\partial y} u(x,y)( y - xy^2 ) = \frac{\partial}{\partial x} u(x,y)(x + x^2y^2) $$
(The two sides must be equal for u(x,y) to make the original equation exact, since $ \frac{\partial M(x,y)}{\partial y} = \frac{\partial N(x,y)}{\partial x} $ for an exact equation)
3. Now you have:$$ u_y(x,y)(y-xy^2) + u(x,y)(1-2xy) = u_x(x,y)(x+x^2y^2) + u(x,y)(1+2xy^2) $$
I'm kind of stuck now. What is the next step, or is this an incorrect start?