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If $\mathrm{y}=\sin (\ell \mathrm{nx})$, then which one of the following is correct?
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Verified Answer
The correct answer is:
$\mathrm{x}^{2} \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}+\mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}=0$
(c) $y=\sin (\log x)$
$\Rightarrow \frac{d y}{d x}=\frac{\cos (\log x)}{x}$
$\Rightarrow \quad x\left(\frac{d y}{d x}\right)=\cos (\log x)$
Again differentiating, $x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}=-\frac{\sin \log x}{x}=\frac{-y}{x}$
$\Rightarrow \quad x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y=0$
$\Rightarrow \frac{d y}{d x}=\frac{\cos (\log x)}{x}$
$\Rightarrow \quad x\left(\frac{d y}{d x}\right)=\cos (\log x)$
Again differentiating, $x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}=-\frac{\sin \log x}{x}=\frac{-y}{x}$
$\Rightarrow \quad x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y=0$
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