Let y = y(x) be the solution of the differential equation xdy = (y...

We have, $xdy = \left(y + {x}^{3}\cos x\right)dx$

$⇒ xdy = ydx + {x}^{3}cosxdx$

$⇒ \dfrac{xdy - ydx}{{x}^{2}} = \dfrac{{x}^{3}cosxdx}{{x}^{2}}$

$⇒ \dfrac{d}{dx}\left(\dfrac{y}{x}\right) = ∫ xcosxdx$

$⇒ \dfrac{y}{x} = xsinx - ∫ 1 . sinxdx$

Therefore, $\dfrac{y}{x} = xsinx + \cos x + C$

At $x = \pi, y = 0,$

$0 = - 1 + C$

$⇒ C = 1, x = \pi, y = 0$

So, $\dfrac{y}{x} = xsinx + \cos x + 1$

$⇒ y = {x}^{2}\sin x + xcosx + x$

Hence, $y\left(\dfrac{\pi}{2}\right) = \dfrac{{\pi}^{2}}{4} + \dfrac{\pi}{2} .$