<s>[INST] Solution to the differential equation $(2x^2+xy-2y^2)dx+(3x^2+2xy)dy=0$ Question: Find the solution to the differential equation $(2x^2+xy-2y^2)dx+(3x^2+2xy)dy=0$
The question prompts me to let $G=(2x^2+xy-2y^2)dx+(3x^2+2xy)dy$ and prove that $e^\frac{y}{x}\frac{G}{x}$ is an exact differential. But what is an exact differential anyways? Is it to multiply $e^\frac{y}{x}$ into $\frac{G}{x}$ and differentiate $e^\frac{y}{x}(3x^2+2xy)$ by $x$ and $e^\frac{y}{x}(2x^2+xy-2y^2)$ by $y$ and prove that both functions are the same?
I'm stuck on this question and would appreciate some help. Thanks!
 [/INST] Hint.
Note that
$$
\frac{\partial}{\partial y}\left(\frac{e^{\frac{y}{x}} \left(2 x^2+x y-2 y^2\right)}{x}\right) = \frac{e^{\frac{y}{x}} \left(3 x^2-3 x y-2 y^2\right)}{x^2}\\
\frac{\partial}{\partial x}\left(\frac{e^{\frac{y}{x}} \left(3 x^2+2 x y\right)}{x}\right) = \frac{e^{\frac{y}{x}} \left(3 x^2-3 x y-2 y^2\right)}{x^2}
$$
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