Question

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

Answer 1

$\underline{\bold{Solution:-}}$

$y = {sin}^{ - 1} x \\ \\ on \: differentiating \: both \: sides\\ \\ \frac{dy}{dx} = \frac{1}{ \sqrt{1 - {x}^{2} } } \\ \\ ( \sqrt{1 - {x}^{2} }) \frac{dy}{dx} = 1 \\ \\ on \: diffentiating \: both \: sides \\ \\ \sqrt{1 - {x}^{2} } \frac{ {d}^{2} y}{ {dx}^{2} } + \frac{dy}{dx} . \frac{d}{dx} ( \sqrt{1 - {x}^{2} } ) = 0 \\ \\ \sqrt{1 - {x}^{2} } \frac{ {d}^{2} y}{ {dx}^{2} } + \frac{dy}{dx} \frac{1 }{2 \sqrt{1 - {x}^{2} } } \frac{d}{dx} (1 - {x}^{2} ) = 0 \\ \\ \sqrt{1 - {x}^{2} } \frac{ {d}^{2} y}{ {dx}^{2} } + \frac{dy}{dx} \frac{1 }{2 \sqrt{1 - {x}^{2} } } ( - 2x) = 0 \\ \\ on \: multiplying \: both \: sides \: by \: \sqrt{1 - {x}^{2} } \\ \\ (1 - {x}^{2} ) \frac{ {d}^{2} y}{ {dx}^{2} } -x { (\frac{dy}{dx} )}^{2} = 0 \\ \\ (1 - {x}^{2} )y2 - xy1 = 0$


Hence proved