Given (-2ysinx+2xy^3-e^xy^2)dx+(4cosx+4x^2y^2-3e^xy)dy=0, y(0)=2

Solve.

>What I did so far was simple enough. The equation is not homogenous, non-seperable and not exact. So logic dictates I should use an integrating factor 'u'. Since our class is a 2nd year ODE course, we use a special case where the factor u is a single variable function of x or y.

Thus,
My=-2sinx+2xy^2-e^xy
Nx=
-2sinx+6xy^2-2e^xy

so

u'/u= (My-Nx)/M
>
du/u
=(-2sinx+2xy^2-e^xy)/(-2sinx+6xy^2-2e^xy)dx

I need to solve for u(y) in this case. No idea what to do. Any help greatly appreciated