(Solved): Use the Laplace transform, inverse Laplace transform, and the Cauchy Residue Theorem to find the s ...

Use the Laplace transform, inverse Laplace transform, and the Cauchy Residue Theorem to find the solution \( x(t) \) for the following equation of motion: \[ m \ddot{x}(t)+b \dot{x}(t)+k x(t)=10, \] where \( m=1, b=4, k=5, x(0)=1 \), and \( \dot{x}(0)=0 \). Using the solution for \( x(t) \) found in Problem 9 , find the time \( \left(t_{\max }\right) \) at which the maximum displacement \( x_{\max } \) occurs and find the value of \( x_{\max } \).