Question

If $y=\sin \left(m \sin ^{-1} x\right)$, then $\left(1-x^{2}\right) y_{2}-x y_{1}$ is equal to $\left(\right.$ Here, $y_{n}$ denotes $\left.\frac{d^{n} y}{d x^{n}}\right)$

• A. $m^{2} y$
• B. $-m^{2} y$
• C. $2 m^{2} y$
• D. $-2 m^{2} y$

Answer 1

Answer: (B)

$y=\sin \left(m \sin ^{-1} x\right)$
$y_{1}=\cos \left(m \sin ^{-1} x\right) \cdot m \cdot \frac{1}{\sqrt{1-x^{2}}}$
where $\left(y_{1}=\frac{d y}{d x}\right)$
$y_{1} \sqrt{1-x^{2}}=m \cos \left(m \sin ^{-1} x\right)$
$\Rightarrow y_{2} \sqrt{1-x^{2}}+ y_{1} \frac{1}{2 \sqrt{1-x^{2}}} \cdot(-2 x)$
$=-m \sin \left(m \sin ^{-1} x\right) \cdot \frac{m}{\sqrt{1-x^{2}}}$
$\left(\because y_{2}=\frac{d^{2} y}{d x^{2}}\right)$
$\Rightarrow y_{2} \sqrt{1-x^{2}}-\frac{x y_{1}}{\sqrt{1-x^{2}}}$
$=\frac{-m^{2}}{\sqrt{1-x^{2}}} \sin \left(m \sin ^{-1} x\right)$
$\Rightarrow y_{2}\left(1-x^{2}\right)-x y_{1}=-m^{2} y$