Let y = y(x) be the solution of the differential equation (1 - x ^...

Given,

$\left(1 - {x}^{2}\right)\dfrac{dy}{dx} = xy + \left({x}^{3} + 2\right)\sqrt{1 - {x}^{2}}$

$⇒ \dfrac{dy}{dx} + \left(\dfrac{- x}{1 - {x}^{2}}\right)y = \dfrac{{x}^{3} + 2}{\sqrt{1 - {x}^{2}}}$

$IF = {e}^{∫ \dfrac{- x}{1 - {x}^{2}}dx} = \sqrt{1 - {x}^{2}}$

$y\left(x\right) \cdot \sqrt{1 - {x}^{2}} = \dfrac{{x}^{4}}{4} + 2x + c$

$y\left(0\right) = 0 ⇒ c = 0$

$\sqrt{1 - {x}^{2}}y\left(x\right) = \dfrac{{x}^{4}}{4} + 2x$

So,required value $= {∫}_{- \dfrac{1}{2}}^{\dfrac{1}{2}}\left(\dfrac{{x}^{4}}{4} + 2x\right)dx - \dfrac{1}{4} \cdot 2{∫}_{0}^{\dfrac{1}{2}}{x}^{4}dx$

$= \dfrac{1}{10}{\left({x}^{5}\right)}_{0}^{\dfrac{1}{2}} = \dfrac{1}{320}$

${k}^{- 1} = 320$