Question

If y = a sin mx + b cos mx, then $\frac{d^{2}y}{d{x}^2}$ is equal to

(a) −m²y
(b) m²y
(c) −my
(d) my

Answer 1

(a) −m²y

Here,

$$\begin{gathered} y=a\sin mx+b\cos mx \\ \Rightarrow \frac{\mathit{d}y}{\mathit{d}x}=am\cos mx-bm\sin mx \\ \Rightarrow \frac{{\mathit{d}}^{\mathit{2}}y}{\mathit{d}{x}^{\mathit{2}}}=-a{m}^2\sin mx-b{m}^2\cos mx \\ =-m^2\left(a\sin mx+b\cos mx\right) \\ =-m^{2}y \end{gathered}$$