If the solve the problem

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:

$y=a \cos n t-b \sin n t$

$\frac{d y}{d t}=-a n \sin n t-b n \cos n t$

$\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dt}^{2}}=-\mathrm{an}^{2} \cos \mathrm{n} \mathrm{t}+\mathrm{bn}^{2} \sin \mathrm{n} \mathrm{t}=\lambda \mathrm{y}$

$\lambda y=-n^{2}(a \cos n t-b \sin n t)$

$\lambda y=-n^{2} y$

$\lambda=-n^{2}$

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