Consider a forced damped harmonic oscillator:
\[ \ddot { y } ( t ) + 2 k \dot { y } ( t ) + \Omega ^ { 2 } y ( t ) = f ( t )\tag{25} \]
the Green function G(t) is defined such that
\[ y ( t ) = \int d t ^ { \prime } G ( t - t ^ { \prime } ) f ( t ^ { \prime } ) .\tag{26} \]
Find G(t) when
\(\Omega > k > 0\) (oscillating system)
\(\Omega = k\) (critical damping)
\(k > \Omega > 0\) (overdamped system)