I'm trying to analyze this mass-spring system -- i.e. write down the differential equation governing it.
As you can see, there is a block of mass $m_1$ attached to a wall by an ideal spring of spring constant $k_1$. On the other side the block is attached by another ideal spring, of spring constant $k_2$, to a motor which is driving the system with arbitrary force $F(t)$.
I'm really just not sure how to account for the motor. Whenever it pulls (or pushes) on the $k_2$ spring, both the $k_2$ and $k_1$ springs expand or contract, but not by the same amount.
If the motor weren't there, the equation for a change in position of the block should be $\sum F = -k_1x -k_2x \implies m\ddot x + (k_1 + k_2)x =0$.
However with the motor the change in position is dictated by the function $F(t)$ and I'm not sure how to account for that. Will it just be that the motor will add an additional force to the spring -- i.e. $m\ddot x + (k_1 + k_2)x -F(t) =0$? That doesn't seem right because it doesn't take into account that the stretching of the springs is entirely due to $F(t)$.
I'm just not sure how to set up this problem. How should I be thinking about this?