Question

Solve the initial value problem. x y' = y + 7 x^2 sin\(x\), y(4 pi) = 0

Answer 1

$xy'=y+7x^2\sin x$
$xy'-y=7x^2\sin x$
$\dfrac1xy'-\dfrac1{x^2}y=7\sin x$
$\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac1xy\right]=7\sin x$
$\dfrac1xy=\displaystyle\int7\sin x\,\mathrm dx$
$\dfrac1xy=-7\cos x+C$
$y=-7x\cos x+Cx$

With the initial value $y(4\pi)=0$, we have

$0=-28\pi+4\piC\implies C=7$

so that the particular solution to the ODE is

$y=-7x\cos x+7x$