\(\displaystyle \frac{dy}{dx} = \frac{xy^2 - \cos x \sin x}{y(1 - x^2)}\)
\(\displaystyle (xy^2 - \cos x \sin x) \ dx + y(x^2 - 1) \ dy = 0\)
\(\displaystyle M = xy^2 - \cos x \sin x\)
\(\displaystyle N = y(x^2 - 1)\)
\(\displaystyle \frac{\partial M}{\partial y} = 2xy = \frac{\partial N}{\partial x}\)
\(\displaystyle \frac{\partial f}{\partial x} = M = xy^2 - \cos x \sin x\)
\(\displaystyle \int \frac{\partial f}{\partial x} \partial x = \int (xy^2 - \cos x \sin x) \ dx\)
\(\displaystyle f(x,y) = \frac{x^2y^2}{2} + \frac{\cos^2 x}{2} + g(y)\)
\(\displaystyle \frac{\partial f}{\partial y} = N = y(x^2 - 1) = x^2y + g'(y)\)
\(\displaystyle g'(y) = y(x^2 - 1) - x^2y\)
\(\displaystyle \frac{dg}{dy} = y(x^2 - 1) - x^2y\)
\(\displaystyle dg = (y(x^2 - 1) - x^2y) \ dy\)
\(\displaystyle \int dg = \int (y(x^2 - 1) - x^2y) \ dy\)
\(\displaystyle g(y) = \frac{y^2(x^2 - 1)}{2} - \frac{x^2y^2}{2}\)
\(\displaystyle f(x,y) = \frac{x^2y^2}{2} + \frac{\cos^2 x}{2} + \frac{y^2(x^2 - 1)}{2} - \frac{x^2y^2}{2} = \frac{\cos^2 x}{2} + \frac{y^2(x^2 - 1)}{2}\)
Then the solution to the differential equation is:
\(\displaystyle \frac{\cos^2 x}{2} + \frac{y^2(x^2 - 1)}{2} = D\)
Or
\(\displaystyle \cos^2 x + y^2(x^2 - 1) = 2D\)
Or
\(\displaystyle \cos^2 x + y^2(x^2 - 1) = C\)
