Linear First Order ODE/y' - (y over x) = 3 x

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Theorem

The linear first order ODE:

$(1): \quad \dfrac {\d y} {\d x} - \dfrac y x = 3 x$

has the general solution:

$\dfrac y x = 3 x + C$

or:

$y = 3 x^2 + C x$


Proof

This is an instance of Linear First Order ODE: $y' - \dfrac y x = k x$.

Its solution is:

$\dfrac y x = k x + C$

or:

$y = k x^2 + C x$

from which the solution to $(1)$ is found by substituting $3$ for $k$.

$\blacksquare$