Explanation

(a)
Given, $y = \left( x + \sqrt{1 + x^{2}} \right)^{n}$
$\Rightarrow$ $\frac{dy}{dx} = n\left\lbrack x + \sqrt{1 + x^{2}} \right\rbrack^{n - 1}\left( 1 + \frac{x}{\sqrt{x^{2} + 1}} \right)$
$= \frac{n\left\lbrack x + \sqrt{1 + x^{2}} \right\rbrack^{n}}{\sqrt{1 + x^{2}}}$
$\Rightarrow$ $\left( \frac{dy}{dx} \right)^{2}\left( 1 + x^{2} \right) = n^{2}y^{2}$
Again, differentiating, we get
$2\ \frac{dy}{dx}.\frac{d^{2}y}{dx^{2}}\left( 1 + x^{2} \right) + 2x\left( \frac{dy}{dx} \right)^{2} = 2n^{2}y\ \frac{dy}{dx}$
$\Rightarrow$ $\frac{d^{2}y}{dx^{2}}\left( 1 + x^{2} \right) + x\frac{dy}{dx} = n^{2}y$ [divide by $2\frac{dy}{dx}$]