Question

If y = 2 sin x + 3 cos x, show that $\frac{d^{2}y}{d{x}^2}+y=0$.

Answer 1

Here,
$$\begin{gathered} y=2\sin x+3\cos x \\ \mathrm{Differentiating}\mathrm{w}.\mathrm{r}.\mathrm{t}.\mathit{}x,\mathrm{we}\mathrm{get} \\ \frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}=2\cos x-3\sin x \\ \mathrm{Differentiating}\mathrm{again}\mathrm{w}.\mathrm{r}.\mathrm{t}.\mathit{}x,\mathrm{we}\mathrm{get} \\ \frac{{\mathit{d}}^{\mathit{2}}\mathit{y}}{\mathit{d}{\mathit{x}}^{\mathit{2}}}=-2\mathrm{sinx}-3\mathrm{cosx} \\ \Rightarrow \frac{{\mathit{d}}^{\mathit{2}}y}{\mathit{d}{x}^{\mathit{2}}}=-\left(2\sin \mathrm{x}+3\cos \mathrm{x}\right) \\ \Rightarrow \frac{{\mathit{d}}^{\mathit{2}}y}{\mathit{d}{x}^{\mathit{2}}}=-\mathrm{y} \\ \Rightarrow \frac{{\mathit{d}}^{2}y}{\mathit{d}{x}^2}+\mathrm{y}=0 \end{gathered}$$

Hence proved.