Question

If x = a cos nt − b sin nt and $\frac{d^{2}x}{dt}=\mathrm{\lambda }x$, then find the value of λ.

Answer 1

Here,

$$\begin{gathered} x=a\cos nt-b\sin nt \\ \mathrm{Now}, \\ \frac{\mathit{d}x}{\mathit{d}t}=-an\sin nt-bn\cos nt \\ \frac{{\mathit{d}}^{\mathit{2}}x}{\mathit{d}{t}^{\mathit{2}}}=-a{n}^2\cos nt+b{n}^2\sin nt \\ \mathrm{Also}, \\ \frac{{\mathit{d}}^{\mathit{2}}x}{\mathit{d}{t}^{\mathit{2}}}=\lambda x\left[\mathrm{Given}\right] \\ \Rightarrow -a{n}^2\cos nt+b{n}^2\sin nt=\lambda \left(a\cos nt-b\sin nt\right) \\ \Rightarrow -n^2\left(a\cos nt-b\sin nt\right)=\lambda \left(a\cos nt-b\sin nt\right) \\ \Rightarrow \lambda =-n^2 \end{gathered}$$