Q242 / 270 · NDA 2018 (Phase 2) · Mathematics

If $y=\sin (\ln x)$, then which one of the following is correct?

  • A
    $\frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{dx}^2}+\mathrm{y}=0$
  • B
    $\frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{dx}^2}=0$
  • Correct answer
    C
    $x^2 \frac{d^2 y}{d x^2}+x \frac{d y}{d x}+y=0$
  • D
    $x^2 \frac{d^2 y}{d x^2}-x \frac{d y}{d x}+y=0$

Solution

$\begin{aligned} & \text { (c) } y=\sin (\log x) \\ & \Rightarrow \quad \frac{d y}{d x}=\frac{\cos (\log x)}{x} \\ & \Rightarrow \quad x\left(\frac{d y}{d x}\right)=\cos (\log x) \end{aligned}$ Again differentiating, $\begin{aligned} & x \frac{d^2 y}{d x^2}+\frac{d y}{d x}=-\frac{\sin \log x}{x}=\frac{-y}{x} \\ \Rightarrow \quad & x^2 \frac{d^2 y}{d x^2}+x \frac{d y}{d x}+y=0 \end{aligned}$

Hence, the correct answer is option: 'C'.

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