Response of First-Order Circuit with Constant Input Confusion

I don't have any question in particular, the source of my confusion is more conceptual.

First lets consider a regular RC circuit with a switch that asks us to find, say Vc(t) for t>0 (Let's assume the switch will close or open at t=0)

So I can find Vc(0-) (i.e. the time interval right before 0).

After this I'm going to attempt to find Vc(t) for the interval t>0.

Now here is where I get confused:

When do I know if I can apply the following equation,



?

In other words, in what cases can I not use the following equation?

Can I use this equation when dealing with a unit step source?

 

You couldn't use it in the case where the input changes within the interval 0<t<∞. For instance when the input is a pulse from say [0 <= t <= T]. The equation only provides for a zero or constant input over the aforementioned time range with the inclusion of an initial condition for the capacitor voltage.

You can use it for a unit step input.

 

So essentially I can use in a given time interval T1 < t < T2 if the input is CONSTANT, correct?

If it were a step source that has a sin or a exponential function in it, then I would not be able to apply that equation, correct?