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Q. The differential equation for unforced under-damped oscillatory motion can be written [tex]\mathop x\limits^{..} + 2p\mathop x\limits^. + \omega ^2 x = 0[/tex] where the constants p and omega satisfy [tex]0 < p < \omega [/tex].

Find the general solution of this differential equation, and show that this solution can be expressed in the form [tex]x = {\mathop{\rm Re}\nolimits} ^{ - pt} \cos \left( {\sqrt {\omega ^2 - p^2 } t - \alpha } \right)[/tex] where R and alpha are constants.

[tex]

x\left( t \right) = e^{ - pt} \left( {c_1 \cos \left( {t\sqrt {p^2 - \omega ^2 } } \right) + c_2 \sin \left( {t\sqrt {p^2 - \omega ^2 } } \right)} \right)

[/tex] by using 0 < p < omega.

As can be seen I haven't been able to get too far. I can't think of a way to get to the answer. Can someone please help me out?