Exercise 4

\[x(t)=A \cos \omega t + B \sin \omega t\] \[\dot{x}(t) = A \omega (-\sin \omega t) + B \omega \cos \omega t\]

\[\ddot{x}(t) = - \omega^2 A \cos \omega t - \omega^2 B \sin \omega t\]

\[\ddot{x}(t) = - \omega^2 (A \cos \omega t + B \sin \omega t) = - \omega^2 x(t)\]

Initial conditions for \(t=0\):

\[x_0 = x(0) = A\]

\[v_0 = \dot{x}(0) = B \omega \Rightarrow B = \frac{v_0}{\omega}\]