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