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

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

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

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

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

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

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