y = log |x + $\sqrt{x^2+a^2}$| ... (1) Differentiating (1) w.r.t. x, we get $\frac{dy}{dx}$ = $\frac{1+\frac{x}{\sqrt{x^2+a^2}}}{x+\sqrt{x^2+a^2}}$ ⇒ $\frac{dy}{dx}$ = $\frac{1}{\sqrt{x^2+a^2}}$ ... (2) ⇒ x $\frac{dy}{dx}$ = $\frac{x}{\sqrt{x^2+a^2}}$ ... (3) Again, differentiating (2) w.r.t. x, we get ⇒ $\frac{d^2y}{dx^2}$ = $\frac{-\frac{2x}{(2x^2+a^2)^{1/2}}}{x^2+a^2}$ ⇒ $\frac{d^2y}{dx^2}$ = - $\frac{x}{(x^2+a^2)^{3/2}}$ ⇒ $x^2+a^2 \frac{d^2y}{dx^2}$ = - $\frac{x}{\sqrt{x^2+a^2}}$ ... (4) Adding equation (3) and (4), we get $x^2+a^2 \frac{d^2y}{dx^2}$ + x $\frac{dy}{dx}$ = - $\frac{x}{\sqrt{x^2+a^2}}$ + $\frac{x}{\sqrt{x^2+a^2}}$ = 0 ⇒ $x^2+a^2 \frac{d^2y}{dx^2}$ + x $\frac{dy}{dx}$ = 0