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

Theorem

The linear first order ODE:

$\dfrac {\d y} {\d x} + \dfrac y x = 3 x$

has the general solution:

$x y = x^3 + C$

or:

$y = x^2 + \dfrac C x$

Proof

This is a special case of:

Linear First Order ODE: $\dfrac {\d y} {\d x} + \dfrac y x = k x^n$

where $k = 3$ and $n = 1$, yielding:

$y = x^2 + \dfrac C x$

Multiplying through by $x$ reveals:

$x y = x^3 + C$

$\blacksquare$

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 2.10$: Example $1$