Solve the equations of motion using the Laplace transform approach:

(a) \(\ddot{y}+2 \dot{y}+3 y=5 \cos 3 t, y(0)=3, \dot{y}(0)=4\)

(b) \(4 \ddot{y}+5 \dot{y}+5 y=4 u(t), y(0)=1, \dot{y}(0)=1\), where \(u(t)\) is the unit step function

(c) \(3 \ddot{y}+3 \dot{y}+6 y=3 e^{-t}+2 \cos 3 t, y(0)=2, \dot{y}(0)=4\)

(d) \(\ddot{y}+\dot{y}+y=F(t), y(0)=0, \dot{y}(0)=0\), where \(F(t)\) is given by a square wave function with maximum amplitude 1 and period 1

(e) \(\ddot{y}+2 \dot{y}+3 y=\cos 3 t+\cos 5 t, y(0)=0, \dot{y}(0)=0\).