082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

554 Chapter 19 Ordinary differential equations I
Then
ie Rt/L = 1 L
= 1 L
∫
te Rt/L dt
( te
Rt/L
R/L − ∫ e
Rt/L
R/L dt )
= teRt/L
R − L R 2 eRt/L +K
whereK isthe constant of integration. The general solution is
i = t R − L R 2 +Ke−Rt/L
When t = 0, i = 0. This gives the initial condition required to find K. Applying
i(0) = 0 givesK =L/R 2 so thatthe particular solution is
i = t R + L R 2 (
e −Rt/L −1 )
In many engineering applications the terms transient response, steady-state response,
zero-input response, and zero-state response arise through the solution of differential
equations.These termsare explained inthe following example.
Engineeringapplication19.4
RC circuit:zero-inputresponseandzero-stateresponse
The differential equation which is used to model the variation in voltage across a
capacitor, v C
(t), inaseriesRC circuitwasderived inEngineeringapplication 19.1:
RC dv C
+v
dt C
=v S
(t)
where v S
(t) is the applied voltage. Suppose that this applied voltage takes the form
v S
(t) = V cos ωt. Suppose also that when the switch is closed att = 0, the initial
voltage across the capacitor isV 0
, thatis v C
(0) =V 0
.
(a) Show that this equation can be written in the form of Equation (19.4); that is, it
is a first-order linear equation.
(b) Usetheintegratingfactormethodtoobtaintheparticularsolutionofthisequation
which satisfies the given initial condition.
(c) Obtain the particular solution of the equation subject instead to the initial condition
v C
(0) = 0. It corresponds to the response of the system when there is no
initialenergyinthecircuit.Thissolutionisreferredtoasthezero-stateresponse.
(d) Obtainthesolutionoftheequationinthecasewhenthesupplyvoltageisidenticallyzero,
v S
(t) = 0,subjecttothegivencondition v C
(0) =V 0
.Thissolutionis
often referred to as the zero-input response, and represents the response of the
systemwhen the input iszero.