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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} …