<s>[INST] Using integrating factors to solve difficult differential equations Consider the differential equatioN:
$$(2x^2 y -2y^4)dx+ ( 2x^3 + 3xy^3) dy = 0$$
This equation is of the form:
$$ Q dx +  P dy=0$$
Now,  it's easy to see that this differential is not exact by using the commutativity of second order partial condition. Multiply it by an integrating factor $\eta(x,y)$ such that it does:
$$ \eta_x  P + P_x \eta  = \eta_y Q + Q_y \eta$$
Or,
$$ \eta_x P - \eta_y Q + ( P_x - Q_y) \eta = 0$$
This becomes:(*)
$$ \eta_x ( 2x^3 + 3xy^3) - \eta_y ( 2x^2 y - 2y^4) + ( 6x^2 +3 y^3 - 2x^2 -8y^3) \eta =0$$
Or,
$$ \eta_x ( 2x^3 + 3xy^3) - \eta_y ( 2x^2 y - 2y^4) + ( 4x^2  - 11y^3) \eta =0$$
Now... I'm not quite sure what to do.. I don't think I can write $ \eta$ as function solely of $x$ or $y$.. did I miss something or... ?

As correctly observed  by @Aleksas Domarkas. There is a mistake after the place where I put *. The current working is as follows:
$$ \eta_x ( 2x^3 + 3xy^3) - \eta_y ( 2x^2 y - 2y^4) + ( 6x^2 +3 y^3 - 2x^2 +8y^3) \eta =0$$
Or,
$$ \eta_x ( 2x^3 + 3xy^3) - \eta_y ( 2x^2 y - 2y^4) + ( 4x^2  + 11y^3) \eta =0$$
 [/INST] I get ("+" in last term).
$$\eta_x ( 2x^3 + 3xy^3) - \eta_y ( 2x^2 y - 2y^4) + ( 4x^2  + 11y^3) \eta =0$$
Let $\eta=x^ay^b$. Then
$$\left( -2 b-3 a-11\right) \, {{x}^{a}}\, {{y}^{b+3}}+\left( 2 b-2 a-4\right) \, {{x}^{a+2}}\, {{y}^{b}}=0$$
Solve system $-2 b-3 a-11=0,\quad 2 b-2 a-4=0$.
Then $a=-3,\quad b=-1$. Integrating factor is
$$\eta=\frac{1}{x^3y}$$
We get general solution of differential equation
$$2\ln(xy)+\frac{x^3}{y^2}=C$$
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