Linear First Order ODE/x dy + y dx = x cosine x dx/Proof 2

Theorem

The linear first order ODE:

$x \rd y + y \rd x = x \cos x \rd x$

has the general solution:

$x y = x \sin x + \cos x + C$

Proof

\(\ds x \dfrac {\d y} {\d x} + y\)

\(=\)

\(\ds x \cos x\)

\(\ds \leadsto \ \ \)

\(\ds x y\)

\(=\)

\(\ds \int x \cos x \rd x + C\)

Linear First Order ODE: $x y' + y = \map f x$

\(\ds \)

\(=\)

\(\ds x \sin x + \cos x + C\)

Primitive of $x \cos a x$

$\blacksquare$