Question

$Ify=e^{aco{s}^{-1}x},-1\le x\le 1showthat(1-x^2)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}-a^{2}y=0$

Answer 1

Given, $y=e^{aco{s}^{-1}x}$ .....(i)

Differentiating w.r.t. x, we get

$\Rightarrow \frac{dy}{dx}=e^{aco{s}^{-1}x}\frac{d}{dx}aco{s}^{-1}x=e^{aco{s}^{-1}x}.\frac{-a}{\sqrt{1-x^2}}\Rightarrow \frac{dy}{dx}=\frac{-ay}{\sqrt{1-x^2}}(\because e^{aco{s}^{-1}x}=y)\Rightarrow \sqrt{1-x^2}\frac{dy}{dx}=-ay....\left(ii\right)Againdifferentiatingw.r.t.x,weget\sqrt{1-x^2}\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}.\frac{-2x}{2\sqrt{1-x^2}}=-a\frac{dy}{dx}\Rightarrow \sqrt{1-x^2}\frac{d^{2}y}{d{x}^2}+\left(\frac{-x}{\sqrt{1-x^2}}\right)\frac{dy}{dx}=\frac{-a(-ay)}{\sqrt{1-x^2}}(UsingEq.(ii\left)\right)Multiplyingthroughoutby\sqrt{1-x^2},weget(1-x^2)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}-a^{2}y=0$