Question

If y = sin^-1x show that (1-x²) d²y/dx² -x dy/dx =0.

Answer 1

y = sin^-1x

∴ \(\frac{dy}{dx}=\frac1{\sqrt{1-x^2}}\) = (1 - x²)^-1/2

\(\frac{d^2y}{dx^2}=\frac12(1-x^2)^{-3/2}\times-2x\)

 = \(\frac{x}{(1-x^2)\sqrt{1-x^2}}\) 

∴ (1 - x²)\(\frac{d^2y}{dx^2}\) = \(\frac{x}{\sqrt{1-x^2}}\) = x\(\frac{dy}{dx}\)

⇒ (1 - x²)\(\frac{d^2y}{dx^2}\) -  x\(\frac{dy}{dx}\) = 0