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,

\(v_{c}(t) = v_{c}(\infty) + ( v_{c}(0) - v_{c}(\infty))e^{\frac{-t}{\tau}}\)

?

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.

 

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?

 

Yes, you can use the equation only for constant input voltages. If they are partially constant (constant for time intervals, that is) you can apply the formula many times for many sequential time intervals.

For non-constant input voltages you must find the transfer function of the RC filter and apply it on the arbitrary input. The transfer function will be found through Kirchoff's Laws and the element equations.