Damped Forced Motion Where forcing happens at a time ahead of 0 So presume that I have the following DE which represents a vibrating spring with a mass attached to it
$$x''+6x'+10x=25\cos(4t), x(0) = \frac{1}{2},x'(0)=0$$
I am lead to believe that the force is applied periodically starting when t = 0 seconds (that is to say the forcing function is applied when the mass is released). What form would this DE have if the force was applied starting when t was some greater positive number? Like say for example, 2 (that is to say the forcing function is applied 2 seconds after the mass is released) ? What form would it have if instead of a periodic forcing function, the forcing function was instead constant like for the DE:
$$x''+6x'+10x=5$$

Case 1: $x''+6x'+10x=25\cos(4t), x(0) = \frac{1}{2},x'(0)=0$. The solution is
$$x(t) = \dfrac{1}{102} e^{-3 t} \left(-172 \sin (t)+100 e^{3 t} \sin (4 t)+76 \cos (t)-25 e^{3 t} \cos (4 t)\right)$$
Plots are


Case 2: $x''+6x'+10x=25\cos(4t), x(2) = \frac{1}{2},x'(2)=0$. The solution is
$x(t) = \dfrac{1}{204}\left(e^{6-3 t} (-306 \sin (2-t)+225 \sin (10-t)-425 \sin (t+6)+102 \cos (2-t)-375 \cos (10-t)+425 \cos (t+6))-50 (\cos (4 t)-4 \sin (4 t)\right)$
Plots are:


Case 3: $x''+6x'+10x=5, x(0) = \frac{1}{2},x'(0)=0$. The solution is
$$x(t) = \dfrac{1}{2}$$
The plot is

