Linear First Order ODE/(2 y - x^3) dx = x dy

Theorem
The linear first order ODE:
 * $(1): \quad \left({2 y - x^3}\right) \mathrm d x = x \, \mathrm d y$

has the solution:
 * $y = -x^3 + C x^2$

Proof
Rearranging $(1)$:
 * $(2): \quad \dfrac {\mathrm d y} {\mathrm d x} - \dfrac 2 x y = - x^2$

$(2)$ is in the form:
 * $\dfrac {\mathrm d y}{\mathrm d x} + P \left({x}\right) y = Q \left({x}\right)$

where $P \left({x}\right) = -\dfrac 2 x$.

Thus:

Thus from Solution by Integrating Factor, $(1)$ can be rewritten as:
 * $\dfrac {\mathrm d} {\mathrm d x} \left({\dfrac y {x^2} }\right) = -1$

and the general solution is:
 * $\dfrac y {x^2} = -x + C$

or:
 * $y = -x^3 + C x^2$