Capacitors store electrostatic energy...E=CV squared÷2.
current-voltage relationship is given by i(t) C×dv÷dt.
For a capacitor a instantaneous changing voltage dt=0 requires an infinite current; "Good Practice" in electrical engineering states, this cannot be achieved.
The change in voltage in a capacitor requires a finite time!
Recognition that the rate of energy produced must be equal to the sum of the rate of energy
dissipated and the rate of energy stored at all times...
(Principle of Energy Conservation).
The equation provided in my post describes the response of a capacitor formulated as linear
time-invariant differential equations with constant coefficients.
The classical method of treating transients, is the use of capacitors
and analytical solutions of linear differential equations.
Transient analysis based upon the laplace transform method is what
I use when analysing transients in a power system.