From my understanding, the zero-input and zero-state responses of an RC circuit can be found by solving for the homogenous and particular solution of the ordinary differential equation found using Kirchhoff's current law. But I am confused as to what happens if there is no resistor in the circuit. In this situation, y(t) = x(t), where x(t) is the voltage input and y(t) is the voltage output across the capacitor. Now, my intuitive understanding tells me that the zero-input response is zero because no current has been provided yet and the capacitor is uncharged. But what about the zero-state response? Since there is no resistor in the circuit, will the capacitor simply not discharge? So, will the zero-state response of the circuit also be zero?
The voltage source has zero internal resistance (not infinite, that's a current source). A capacitor across it will have whatever potential exists at the source's pins. That means that whatever variation of voltage exists, the capacitor will follow exactly, while the current through it will be infinite in value. Make that impossible, since I=V/R, unless you have special cherokee blood.
In short, there is no zero-input since there is a voltage source involved, and zero-state means whatever the source has.