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

550 Chapter 19 Ordinary differential equations I
19.4.3 First-orderlinearequations
First-order linear differential equations can always bewritten inthe ‘standard’ form
dy
+P(x)y =Q(x) (19.4)
dx
wherePandQare both functions of the independent variable,x, only. In some cases,
either of these may be simplyconstants.
An example of suchan equation is
dy
+3xy = 7x2
dx
Comparing this with the standard form in Equation (19.4), note that P(x) = 3x and
Q(x) = 7x 2 . As a second example consider
dy
dx − 2y
x =4e−x
HereP(x) = − 2 x (note inparticular the minus sign) andQ(x) = 4e−x .
Finally, inthe equation
dy
dx −5y=sinx
note thatP(x) issimplythe constant −5.
Variables other thanyandxmay be used. So, forexample,
dx
dt +8tx=3t2 −5t
isafirst-orderlinearequationintheformofEquation(19.4)butwithindependentvariablet
and dependent variablex. HereP(t) = 8t andQ(t) = 3t 2 −5t.
In what follows it will be important that you can distinguish between the dependent
and independent variables, and alsothatyou areable toidentifythe functionsPandQ.
Equations such as these arise naturally when modelling many engineering applications.Forexample,theequationwhichdeterminesthecurrentflowinaseriesRLcircuit
when the applied voltage takes the form of a ramp isgiven by
di
dt + R L i = t L
This is a first-order linear equation in whichP(t) is the constant R L andQ(t) = t L . You
will learnhow tosolve such equations inthe following section.
19.4.4 Theintegratingfactormethod
All first-order linear equations, even when they are not exact, can be made exact by
multiplying them through by a function known as an integrating factor. As we have
seen, the solution then follows by performing an integration. For example, the linear
equation
dy
dx + 3 x y = e2x
x 3
is not exact, but multiplying it through by x 3 produces the differential equation in
Example 19.13(a) which isexact.