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The initial condition of a circuit at rest is, an inductor is considered a short to DC and its current is used as an initial condition, a capacitor is considered a open circuit to DC and its voltage is used as an initial conditions.
Initaial condition is the condition in which the circuit is initially present. A circuit may not be connected initially to a source or initially it was connected to some source. unlike resistor which change in zero time or at once with the circui, capacitor and inductor cannot change their volatge and current respectively instantally.So if iniatially at time at t=0- if the capcitor was disconnected and is completely dicaharged, votage across capcitor is 0 and within the same instant of time it was connectd to some source ,it will not change its voltage but it will maintain 0 volts for t=0+.Similar thing happens for the inductor current.
Whenever a circuit is comprised of energy storing elements like inductor and capacitor is analysed the initial conditions comes into picture. As we know that the voltage and current relationship in capacitor and inductor is a derivative of each other I.e. for inductor v= L di/ DT and for capacitor I= C dv\dt . So for such circuits differential equation represents the mathematical model of the network and the total response of the circuit to a stimulus is a sum of response generated by the source acting alone on a de- energised circuit and response generated by stored energy in capacitors and inductor without stimulus. The circuit is like a memory which stores previously stored charge or current in form of electric field between the plates of a capacitor and magnetic field in a coil. And when this energy vents out through the circuit I response results. Hence initial conditions are very important for determining the solution for circuit with integro- differential equations. For a inductor the current flowing before the application of excitation input at t=0 sec. Wants to flow even after the application of input so current flowing through a coil remains the same just before and after the application of excitation input. Likewise the voltage across a capacitor tries to remain the same after the application of excitation input. But this only exists for a very short time which is called transient as we know that there is only finite energy is stored in the physical elements so the the response dies out soon.