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

Given $xdy - ydx = \sqrt{{x}^{2} - {y}^{2}}dx$

$⇒ \dfrac{xdy - ydx}{{x}^{2}} = \dfrac{1}{x}\sqrt{1 - \dfrac{{y}^{2}}{{x}^{2}}}dx$

$⇒ ∫ \dfrac{d\left(\dfrac{y}{x}\right)}{\sqrt{1 - {\left(\dfrac{y}{x}\right)}^{2}}} = ∫ \dfrac{dx}{x}$

$⇒ \sin^{- 1}\left(\dfrac{y}{x}\right) = \ln \left|x\right| + c$

at $x = 1, y = 0 ⇒ c = 0$

$y = xsin ( \ln x )$

$A = {∫}_{1}^{{e}^{\pi}}xsin\left(\ln x\right)dx$

$x = {e}^{t}, dx = {e}^{t}dt ⇒ {∫}_{0}^{\pi}{e}^{2t}\sin \left(t\right)dt = A$

$\alpha{e}^{2\pi} + \beta = {\left(\dfrac{{e}^{2t}}{5} ( 2\sin t - \cos t )\right)}_{0}^{\pi} = \dfrac{1 + {e}^{2\pi}}{5}$

$\alpha = \dfrac{1}{5}, \beta = \dfrac{1}{5}$

So, $10 ( \alpha + \beta ) = 4$