Question

If y = cot x show that $\frac{d^{2}y}{d{x}^2}+2y\frac{dy}{dx}=0$.

Answer 1

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

Hence proved.