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

19.2 Basic definitions 535
In engineering, differential equations are most commonly used to model dynamic
systems. These are systems which change with time. Examples include an electronic
circuit with time-dependent currents and voltages, a chemical production line in which
pressures, tank levels, flow rates, etc., vary with time, and a semiconductor device in
which hole and electron densities change with time.
19.2 BASICDEFINITIONS
Inordertosolveadifferentialequationitisimportanttoidentifycertainfeatures.Recall
fromChapter2thatinafunctionsuchasy =x 2 +3xwesayxistheindependentvariable
andyis the dependent variable since the value ofydepends upon the choice we have
made forx.
Inadifferential equation suchas
dy
dx −2y=3x2
x isthe independent variable, andyisthe dependent variable.
Similarly, forthe differential equation
dx
dt +7x=et
t isthe independent variable andxisthe dependent variable.
We see thatthe variable being differentiated isthe dependent variable.
Before classifying differential equations, wewill derive one.
Engineeringapplication19.1
AnRC chargingcircuit
Consider the RC circuit of Figure 19.1. Suppose we wish to derive a differential
equation which models the circuit so that we can determine the voltage across the
capacitor at any time,t. Clearly there are two different cases corresponding to the
switchbeingopenandtheswitchbeingclosed.Wewillconcentrateonthelatterand
for convenience assume that the switch is closed att = 0. From Kirchhoff’s voltage
law wehave
v S
=v R
+v C
thatis v R
=v S
−v C
where v S
isthevoltageofthesupply, v C
isthevoltageacrossthecapacitor,and v R
is
the voltageacrossthe resistor.Using Ohm’slaw forthe resistor thengives
i = v S −v C
R
whereiisthecurrentflowinginthecircuitaftertheswitchisclosed.Forthecapacitor
i =C dv C
dt
Combining these equations gives
C dv C
dt
= v S −v C
R
➔