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

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

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

where:
 * $P \left({x}\right) = \dfrac 1 x$
 * $Q \left({x}\right) = \cos x$

Thus:

Thus from Solution by Integrating Factor: