**RC circuit:**When the input to the system represented by the term on the right-hand side of the DE is non-zero, the DE can be solved to find its particular or steady state solution. We first a constant input:

(83) We next consider a sinusoidal input . We first assume the input is a complex exponential input:

(84) (85) (86) where (87) (88) (89) Alternatively, this RC circuit can also be solved more conveniently by the phasor method, if only the steady state solution (the particular solution) is of interest. The phasor of the input voltage is simply 1, and the phasor of the voltage across can be found by voltage divider:

(90) (91) **RL circuit:**Given an RL circuit consisted of a resistor and an inductor in series connected to an AC voltage source , we want to find the current . The governing DE describing the circuit can be obtained based on KVL:

(92) (93) (94) (95) where (96) (97) (98) Alternatively, this RL circuit can also be resolved by the phasor method. Now the current can be much more easily found by generalized Ohm's law. The phasor representation of the input voltage is simply 1, and the overall complex impedance of the two elements in series is:

(99) (100) (101)

In general, the *frequency response function (FRF)* of the system
is defined as the ratio of the output to the input of the system, both
represented as complex exponentials. In this specific case, we have

(102) |

(103) |

(104) |