Case 3] : General solution for under damped system ( $\xi \lt 1$) :

In this case, general solution is given by equation $x = c_1e^{s1t} + c_2e^{s2t}$

Here, $s_1 = (- \xi + j\sqrt{1-\xi ^2}) w_n$

$s_2 = ( -\xi - j \sqrt{1 - \xi ^2}) w_n$

where $j …