If y = e^a cos ^(-1 x), show that (1-x^2) (d^2 y)/(d^2) x-x (d y)/(d x)-a^2 y=0.

$y=e^{a \cos ^{-1} x} \Rightarrow \frac{d y}{d x}=e^{a \cos ^{-1} x} \frac{(-a)}{\sqrt{1-x^{2}}}$<br />Therefore,&nbsp;$\sqrt{1-x^{2}} \frac{d y}{d x}=-a y$&nbsp;... (i)<br />Divide w.r.t.x again, and we get<br />$\sqrt{1-x^{2}} \frac{d^{2} y}{d x^{2}}-\frac{x}{\sqrt{1-x^{2}}} \frac{d y}{d x}=-\frac{a d y}{d x}$<br />$\Rightarrow\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}=-a \sqrt{1-x^{2}} \frac{d y}{d x}$<br />= – a ( -ay ) [ from i ].<br />Hence&nbsp;$\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-a^{2} y=0$