I am playing around with the differential $\mathrm{d}F = (x^2+4xy)\,\mathrm{d}x + (y^4+x^2)\,\mathrm{d}y$. This is not an exact differential, as $$\frac{\partial}{\partial y}(x^2+4xy)=4x$$ and $$\frac{\partial}{\partial x}(y^4+x^2)=2x\neq 4x.$$ The standard procedure is now to find an integrating factor $\mu(x,y)$, such that the differential $\mathrm{d}G=\mu(x,y)(x^2+4xy)\,\mathrm{d}x + \mu(x,y)(y^4+x^2)\,\mathrm{d}y$ becomes exact. This means again that $$\frac{\partial}{\partial y}((x^2+4xy)\mu(x,y))=\frac{\partial}{\partial x}((y^4+x^2)\mu(x,y)).$$ Now that last equation can be simplified by carrying out the derivatives as far as possible. I end up with $$2x\mu(x,x)=(y^4+x^2)\frac{\partial\mu}{\partial x}-(x^2+4xy)\frac{\partial\mu}{\partial y}.$$
That same procedure in lots of cases leads to an equation that is solved by a simple ansatz, such as $\mu$ being a function of $x$ alone, or $y$ alone, or $x+y$ alone, or even a polynomial. However, I have now tried several ideas for an ansatz, but to no avail.
Any suggestions?