Linear First Order ODE/x y' + y = x^2 cosine x/Proof 1

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

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

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

Thus:

Thus from Solution by Integrating Factor:
 * $\dfrac {\mathrm d} {\mathrm d x} \left({x y}\right) = x^2 \cos x$