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

Theorem
The linear first order ODE:
 * $(1): \quad x \, \dfrac {\mathrm d y} {\mathrm d x} + y = f \left({x}\right)$

has the solution:
 * $\displaystyle x y = \int f \left({x}\right) \mathrm d x + C$