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

CBSE Class 12 · 2012 · DifferentiationLong Answer

y = sin^{- 1} x

, show that

(1 - x^{2}) \frac{d ^{2} y}{d x ^{2}} - x \frac{d y}{d x} = 0

Answer: Proved

Differentiate once:

\frac{d y}{d x} = \frac{1}{1 - x ^{2}} \Rightarrow 1 - x^{2} \frac{d y}{d x} = 1

Square to remove the surd:

(1 - x^{2}) (\frac{d y}{d x})^{2} = 1

. Differentiate again:

(1 - x^{2}) \cdot 2 \frac{d y}{d x} \frac{d ^{2} y}{d x ^{2}} + (- 2 x) (\frac{d y}{d x})^{2} = 0

Divide by

2 \frac{d y}{d x}

(nonzero):

(1 - x^{2}) \frac{d ^{2} y}{d x ^{2}} - x \frac{d y}{d x} = 0

CBSE Class 12 Board · 2012 · Subjective