The simple exponential response of the first order differential equation was easy to visualize, but the second order differential equations are more complex in their responses, simply because there are two energy storage elements and their different possible combinations produce varying responses. Before we proceed with a full mathematical development, it would be helpful to create an intuitive feeling about the behavior of such systems in which two energy storage elements are in a loop, such as an inductor and a capacitor: one is capable of storing the current and the other is capable of storing the voltage.

A system as shown in the circuit of Figure 4.11 will serve the purpose for this example. We would like to see the inductive current as the response to the input voltage applied on the capacitor. Let's say the switch S1 on the capacitor C was originally connected to the voltage supply, letting the capacitor store a certain amount of charge. Once the capacitor is fully saturated, we throw the switch towards the inductor L, creating a loop between the inductor and the capacitor. The capacitor starts feeding the current to the inductor and the inductor starts building up the voltage across its terminals. The inductor gains the energy loss from the capacitor, but then the change of voltage across the inductor starts feeding the charge back into the capacitor. The charge gain by the capacitor back from the inductor is seen by the inductor as...