It is the catch all force. If there are some other forces which we decide we need to act on our object we lump them in now and call this good. We classically call F(t) the forcing function.
Putting each of these together provides us the following for Newton's Second Law.
mu′′ = mg - k ( L + u ) - gu′ + F(t)
Or, upon rewriting, we find,
mu′′ + g u′ + ku = mg - kL + F (t )
Now, while the object is at rest in its equilibrium position there are properly two forces acting on the object, the force because of gravity and the force because of the spring. Also, as the object is at rest that is not moving these two forces should be canceling each other out. It means that we should have,
mg = kL
By using this in Newton's Second Law provides us the last version of the differential equation which we'll work with.
mu′′ + g u′ + ku = F (t)
Along with this differential equation we will have the following initial conditions.
u (0) = u0 Initial displacement from the equilibrium position.
u′ (0) = u0′ Initial velocity.
Remember that we'll also be using (1) to find out the spring constant, k. Okay. Here we start looking at some exact cases.