Linear First Order ODE/x dy + y dx = x cosine x dx/Proof 1

Proof
Rearranging:
 * $\dfrac {d y} {\d x} + \dfrac y x = \cos x$

This is a linear first order ODE in the form:
 * $\dfrac {\d y} {\d x} + \map P x y = \map Q x$

where:
 * $\map P x y = \dfrac 1 x$
 * $\map Q x = \cos x$

Thus:

Thus from Solution by Integrating Factor: