Question

If x = a cos nt − b sin nt and \[\frac{d^2 x}{dt} = \lambda x\]  then find the value of λ ?

Answer 1

Here,

\[x = a \cos nt - b \sin nt\]

\[\text { Now,} \]

\[\frac{d x}{d t} = - an \sin nt - bn \cos nt\]

\[ \frac{d^2 x}{d t^2} = - a n^2 \cos nt + b n^2 \sin nt\]

\[\text { Also}, \]

\[\frac{d^2 x}{d t^2} = \lambda x \left[ \text { Given } \right]\]

\[ \Rightarrow - a n^2 \cos nt + b n^2 \sin nt = \lambda\left( a \cos nt - b \sin nt \right)\]

\[ \Rightarrow - n^2 \left( a \cos nt - b \sin nt \right) = \lambda\left( a \cos nt - b \sin nt \right)\]

\[ \Rightarrow \lambda = - n^2\]