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A simple vibration problem

$u^{\prime\prime}(t) + \omega^2u = 0,\quad u(0)=I,\ u^{\prime}(0)=0,\ t\in (0,T]$

Exact solution: $u(t) = I\cos (\omega t)$ $u(t)$ oscillates with constant amplitude $I$ and (angular) frequency $\omega$ . Period: $P=2\pi/\omega$ .