Linear First Order ODE/x y' + y = x^2 cosine x/Proof 3

Theorem

The linear first order ODE:

$x \, \dfrac {\d y} {\d x} + y = x^2 \cos x$

has the general solution:

$y = 2 \cos x + x \sin x - \dfrac 2 x \sin x + \dfrac C x$

Proof

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

\(=\)

\(\ds x^2 \cos x\)

\(\ds \leadsto \ \ \)

\(\ds x y\)

\(=\)

\(\ds \int x^2 \cos x \d x + C\)

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

\(\ds \)

\(=\)

\(\ds 2 x \cos x + \paren {x^2 - 2} \sin x + C\)

Primitive of $x^2 \cos a x$

\(\ds \leadsto \ \ \)

\(\ds y\)

\(=\)

\(\ds 2 \cos x + x \sin x - \frac 2 x \sin x + \frac C x\)

$\blacksquare$