Case 2] : General solutions of critically damped system ($\xi = 1$) :

In this case both the roots are real and equal.

In this case, the solution is

$x = (c_1 + c_2t)e^{st} = (c_1 e^{st} + c_2te^{st})$

Differentiating

$x = c_1se^{st} + c_2te^{st} + c_2e^{st}$

$= c_1 se^{st} …