A system is considered linear if its output is directly proportional to its input, such as current and voltage relationships in an electrical system (for example, a nonlinear system is a relationship between current and power). The time invariance condition describes a system in which a delay in the input causes the same amount of delay in the output. The solutions that are presented in this chapter require the system to be linear and time invariant.

A linear system has an additive property and a homogeneity property. An additive system is one in which the response to a sum of inputs is the same as the sum of the individual responses, and a system is homogenous when the scaling of the input by some amount also results in the scaling of the output by the same amount (a sinusoidal input remains a sinusoidal output without affecting the frequency of the input signal; only the magnitude and phase may change). It should be noted that we will be dealing only with linear systems, and the differential equations will be linear differential equations.

A system will exhibit a certain response, depending upon the input energy applied to the system. If the response is due to the stored energy such as a charge on a capacitor or current in the inductor, it is the natural response of the system. But if the response is due to some external energy source, it is the forced response