Chapter 7, Problem 139RE

The following problems consider the “beats” that occur when the forcing term of a differential equation causes “slow" and “fast” amplitudes. Consider the general differential equation $a{y}^{″}+by=\cos \left(\omega t\right)$ that governs undamped motion. Assume that $\sqrt{\frac{b}{a}\ne }\omega$ .

139. Assuming the system starts from rest, show that the particular solution can be written as $y=\frac{2}{a\left({\omega }_0{}^2-{\omega }^2\right)}\sin \left(\frac{{\omega }_0-\omega t}{2}\right)\sin \left(\frac{{\omega }_0+\omega t}{2}\right)$ .