Consider the equation of motion of a single-degree-of-freedom system:

\[m \ddot{x}+c \dot{x}+k x=F\]

Derive the condition that leads to divergent oscillations in each of the following cases:

(a) when the forcing function is proportional to the displacement, \(F(t)=F_{0} x(t)\);

(b) when the forcing function is proportional to the velocity, \(F(t)=F_{0} \dot{x}(t)\); and

(c) when the forcing function is proportional to the acceleration, \(F(t)=F_{0} \ddot{x}(t)\).