CAS EXPERIMENT. Undamped Vibrations. (a) Solve the initial value problem y" + Y = cos

Chapter 2, Problem 2.1.176

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CAS EXPERIMENT. Undamped Vibrations. (a) Solve the initial value problem \(y^{\prime \prime}+y=\cos \omega t, \omega^{2} \neq 1, y(0)=0, y^{\prime}(0)=0\). Show that the solution can be written

\(y(t)=\frac{2}{1-\omega^{2}} \sin \left[\frac{1}{2}(1+\omega) t\right] \times \sin \left[\frac{1}{2}(1-\omega) t\right]\)

(b) Experiment with (17) by changing o to see the change of the curves from those for small \(\omega(>0)\) to beats, to resonance and to large values of \(\omega\) (see Fig. 58).

Text Transcription:

y’’+y=cos omega t, omega^2 neq 1, y(0)=0, y’(0)=0

y(t)=2/1-omega^2 sin [1/2(1+omega)t] times sin [1/2 (1-omega)t]

omega(>0)

omega