Question

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

Answer 1

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

Hence proved.