Question

If $y={\sin }^{-1}x$, show that $(1-x^2)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}=0$

Answer 1

$y=si{n}^{-1}x$

Differentiating w.r.t to $x,$ we get

$\Rightarrow \frac{dy}{dx}=\frac{1}{\sqrt{1-x^2}}\Rightarrow \sqrt{1-x^2}\frac{dy}{dx}=1$

Again differentiating w.r.t to $x$, we get

$\sqrt{1-x^2}\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}.\frac{1\times (-2x)}{2\sqrt{1-x^2}}=0$

$\Rightarrow (1-x^2)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}=0$