Linear First Order ODE/y' - (y over x) = k x

Theorem
Let $k \in \R$ be a real number.

The linear first order ODE:
 * $(1): \quad \dfrac {\mathrm d y} {\mathrm d x} - \dfrac y x = k x$

has the solution:
 * $\dfrac y x = k x + C$

or:
 * $y = k x^2 + C x$

Proof
$(1)$ is in the form:
 * $\dfrac {\mathrm d y}{\mathrm d x} + P \left({x}\right) y = Q \left({x}\right)$

where $P \left({x}\right) = -\dfrac 1 x$.

Thus:

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

and the general solution is:
 * $\dfrac y x = k x + C$

or:
 * $y = k x^2 + C x$