If y = [x + x ^ 2 - 1 ] ^ 15 + [x - x ^ 2 - 1 ] ^ 15 , then ( x ^ 2...

$y = {\left\{x + \sqrt{{x}^{2} - 1}\right\}}^{15} + {\left\{x - \sqrt{{x}^{2} - 1}\right\}}^{15}$

$\dfrac{dy}{dx} = 15{\left\{x + \sqrt{{x}^{2} - 1}\right\}}^{14}\left(1 + \dfrac{x}{\sqrt{{x}^{2} - 1}}\right) + 15{\left\{x - \sqrt{{x}^{2} - 1}\right\}}^{14}\left(1 - \dfrac{x}{\sqrt{{x}^{2} - 1}}\right)$

$\dfrac{dy}{dx} = 15{\left\{x + \sqrt{{x}^{2} - 1}\right\}}^{14}\left(\dfrac{x + \sqrt{{x}^{2} - 1}}{\sqrt{{x}^{2} - 1}}\right) + 15{\left\{x - \sqrt{{x}^{2} - 1}\right\}}^{14}\left(\dfrac{x - \sqrt{{x}^{2} - 1}}{\sqrt{{x}^{2} - 1}}\right)$

$\dfrac{dy}{dx} = \dfrac{15\left[{\left\{x + \sqrt{{x}^{2} - 1}\right\}}^{15} - {\left\{x - \sqrt{{x}^{2} - 1}\right\}}^{15}\right]}{\sqrt{{x}^{2} - 1}}$

$⇒$ $\sqrt{{x}^{2} - 1} . \dfrac{dy}{dx} = 15\left[{\left\{x + \sqrt{{x}^{2} - 1}\right\}}^{15} - {\left\{x - \sqrt{{x}^{2} - 1}\right\}}^{15}\right]$

$⇒$ $\dfrac{x}{\sqrt{{x}^{2} - 1}} . \dfrac{dy}{dx} + \sqrt{{x}^{2} - 1}\dfrac{{d}^{2}y}{d{x}^{2}} = 15 \times \dfrac{15y}{\sqrt{{x}^{2} - 1}}$

$⇒$ $x\dfrac{dy}{dx} + \left({x}^{2} - 1\right)\dfrac{{d}^{2}y}{d{x}^{2}} = 15\sqrt{{x}^{2} - 1} \times \dfrac{15y}{\sqrt{{x}^{2} - 1}} = 225y$