Question:
If $x=a \cos n t-b \sin n t$ and $\frac{d^{2} y}{d t^{2}}=\lambda x$, then find the value of $\lambda$.
Solution:
Given:
$y=a \cos n t-b \sin n t$
$\frac{d y}{d t}=-a n \sin n t-b n \cos n t$
$\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dt}^{2}}=-\mathrm{an}^{2} \cos \mathrm{n} \mathrm{t}+\mathrm{bn}^{2} \sin \mathrm{n} \mathrm{t}=\lambda \mathrm{y}$
$\lambda y=-n^{2}(a \cos n t-b \sin n t)$
$\lambda y=-n^{2} y$
$\lambda=-n^{2}$