Write the correct alternative in the following:

Question:

Write the correct alternative in the following:

If $y=\left(\sin ^{-1} x\right)^{2}$, then $\left(1-x^{2}\right) y_{2}$ is equal to

A. $x y_{1}+2$

B. $x y_{1}-2$

C. $-x y_{1}+2$

D. none of these

Solution:

Given:

$y=\left(\sin ^{-1} x\right)^{2}$

$\frac{d y}{d x}=2 \sin ^{-1} x \frac{1}{\sqrt{1-x^{2}}}$

$\frac{d^{2} y}{d x^{2}}=2\left\{\left(\frac{1}{\sqrt{1-x^{2}}}\right)^{2}+\sin ^{-1} x \frac{\frac{2 x}{2 \sqrt{1-x^{2}}}}{\left(\sqrt{1-x^{2}}\right)^{2}}\right\}$

$=2\left\{\frac{1}{1-x^{2}}+\sin ^{-1} x \frac{x}{\left(\sqrt{1-x^{2}}\right)^{3 / 2}}\right\}$

$\left(1-x^{2}\right) y_{2}=2\left\{1+\sin ^{-1} x \frac{x}{\sqrt{1-x^{2}}}\right\}$

$=2+x\left\{2 \sin ^{-1} x \frac{1}{\sqrt{1-x^{2}}}\right\}$

$=2+x y_{1}$

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