Looks Tough

$\begin{aligned} y & = \left(x + \sqrt{1+x^2}\right)^n \\ \frac {dy}{dx} & = n \left(x + \sqrt{1+x^2}\right)^{n-1} \left(1 + \frac x{\sqrt{1+x^2}}\right) \\ & = \frac {n\left(x + \sqrt{1+x^2}\right)^n}{\sqrt{1+x^2}} \end{aligned}$

$\begin{aligned} \implies \frac {dy}{dx} & = \frac {ny}{\sqrt{1+x^2}} & \small \color{#3D99F6} \text{Multiply both sides by }1+x^2 \\ \left(1+x^2\right) \frac {dy}{dx} & = ny\sqrt{1+x^2} & \small \color{#3D99F6} \text{Differentiate both sides w.r.t. }x \\ \left(1+x^2\right) \frac {d^2y}{dx^2} + 2x \frac {dy}{dx} & = n\sqrt{1+x^2} \frac {dy}{dx} + \frac {nxy}{\sqrt{1+x^2}} \\ & = n\sqrt{1+x^2} \frac {ny}{\sqrt{1+x^2}} + x\frac {dy}{dx} \end{aligned}$

$\begin{aligned} \implies \left(1+x^2\right) \frac {d^2y}{dx^2} + x \frac {dy}{dx} & = \boxed{n^2y} \end{aligned}$