Question

If y = sin (log x), prove that $x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+y=0$.

Answer 1

Here,
$$\begin{gathered} y=\sin \left(\log x\right) \\ \mathrm{Differentiating}\mathrm{w}.\mathrm{r}.\mathrm{t}.x\mathit{,}\mathit{}\mathrm{we}\mathrm{get} \\ \frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}=\frac{\cos \left(\log x\right)}{x} \\ \mathrm{Differentiating}\mathrm{again}\mathrm{w}.\mathrm{r}.\mathrm{t}.x\mathit{,}\mathit{}\mathrm{we}\mathrm{get} \\ \frac{{\mathit{d}}^{\mathit{2}}y}{\mathit{d}{x}^{\mathit{2}}}=\frac{-\sin \left(\log x\right)-\cos \left(\log x\right)}{x^2} \\ \Rightarrow \frac{{\mathit{d}}^{\mathit{2}}\mathit{y}}{\mathit{d}{\mathit{x}}^{\mathit{2}}}=\frac{-\sin \left(\log x\right)}{x^2}-\frac{\cos \left(\log x\right)}{x^2} \\ \Rightarrow \frac{{\mathit{d}}^{\mathit{2}}y}{\mathit{d}{x}^{\mathit{2}}}=\frac{-y}{x^2}-\frac{1}{x}\times \frac{dy}{dx} \\ \Rightarrow x^2\frac{{\mathit{d}}^{\mathit{2}}y}{\mathit{d}{x}^{\mathit{2}}}+x\frac{dy}{dx}+y=0 \end{gathered}$$