I'm attempting to solve a problem involving this differential equation:
$$\frac{dy}{dx} = x^2y^2 + x^2 - y^2 - 1$$
Because this is a separable differential equation, I tend to split the equation between variables $x$ and $y$, and integrate both sites; but I can't seem to get them separated, due to the $x^2y^2$.
$$\frac{dy}{dx} + y^2 = x^2y^2 + x^2 - 1$$ $$ dy + y^2 = x^2y^2\space dx + x^2\space dx - dx$$
Now the only $y$ variable left to get on the opposite side is the $y^2$ that's trapped in $x^2y^2$, but extracting $y^2$ would leave a $y^2$ variable in all the other terms. That's sort of the problem.
$$\frac{dy + y^2}{y^2} = x^2 dx + \frac{x^2}{y^2} dx - \frac{dx}{y^2}$$ $$\frac{dy}{y^2} + 1 = x^2 dx + \frac{x^2}{y^2} dx - \frac{dx}{y^2}$$
And now, if I want the $y$ variables out of the left side, I end up to the previous equation.
$$ dy + y^2 = x^2y^2\space dx + x^2\space dx - dx$$
It's a really bad loop I'm in right now. How can I get around this problem, and get each variable to its seperate side for integration?