<s>[INST] Solution for $y(x^2+y^2)dx + x(3x^2-5y^2)dy = 0$. I am teaching assistant of an elementary course of ODE and a student asked me to solve the following equation:
$$y(x^2+y^2)dx + x(3x^2-5y^2)dy = 0$$
The standard procedure to solve it consists in multiplying the equation by some function $\mu = \mu(x,y)$. Once we do this we must solve
$$-8\mu(x,y)\cdot (x^2-y^2) +\partial_y\mu(x,y)\cdot \left(yx^2 + y^3\right) - \partial_x\mu(x,y)\left(3x^3-5xy^2\right) = 0.$$
In general one assumes that $\mu$ is constant in one of each variable, but this can not be the case here.
I tried lots of manipulations, but could not handle it. I do appreciate any hints.
 [/INST] $$y(x^2+y^2)dx + x(3x^2-5y^2)dy = 0$$
Multiply by $y^2$:
$$y^3x^2dx+y^5dx + x^3dy^3-xdy^5 = 0$$
$$(y^3x^2dx + x^3dy^3)+(y^5dx-xdy^5) = 0$$
Divide by $x^2$:
$$y^3dx + xdy^3-d\left (\dfrac {y^5}x \right) = 0$$
$$d(xy^3)-d\left (\dfrac {y^5}x \right) = 0$$
Integrate:
$$xy^3-\dfrac {y^5}x = K$$
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