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

558 Chapter 19 Ordinary differential equations I
19.5 SECOND-ORDERLINEAREQUATIONS
The general form ofasecond-order linear ordinary differential equation is
p(x) d2 y
dx +q(x)dy +r(x)y = f(x) (19.7)
2 dx
where p(x),q(x),r(x) and f (x)arefunctions ofxonly.
An important relative of thisequation is
p(x) d2 y
dx +q(x)dy +r(x)y = 0 (19.8)
2 dx
whichisobtainedfromEquation(19.7)byignoringthetermwhichisindependentofy.
Equation (19.8) is said to be a homogeneous equation -- all its terms contain y or its
derivatives. Equation (19.7) issaidtobeinhomogeneous.
For example,
x 2d2 y
dx 2 +xdy dx + (x2 −1)y =e −x
isaninhomogeneoussecond-orderlinearequationinwhichp(x) =x 2 ,q(x) =x,r(x) =
x 2 −1 and f (x) = e −x . The associated homogeneous equation is
x d2 y
dx +xdy 2 dx +(x2 −1)y=0
The following properties of linear equations are necessary for finding solutions of
second-order linear equations.
19.5.1 Property1
Ify 1
(x) andy 2
(x) are any two linearly independent solutions of a second-order homogeneous
equation then the general solution,y H
(x), is
y H
(x) =Ay 1
(x) +By 2
(x)
whereA,Bare constants.
We see that the second-order linear ordinary differential equation has two arbitrary
constantsinitsgeneralsolution.Thefunctionsy 1
(x)andy 2
(x)arelinearlyindependent
ifone isnotsimplyamultiple ofthe other.
19.5.2 Property2
Lety P
(x) be any solution of an inhomogeneous equation. Lety H
(x) be the general solution
of the associated homogeneous equation. The general solution of the inhomogeneous
equation isthengiven by
y(x) =y H
(x) +y P
(x)
Inotherwords,tofindthegeneralsolutionofaninhomogeneousequationwemustfirst
findthegeneralsolutionofthecorrespondinghomogeneousproblem,andthenaddtoit
any solutionofthe inhomogeneousequation.
Thefunctiony H
(x)isknownasthecomplementaryfunctionandy P
(x)iscalledthe
particularintegral.Clearlythecomplementaryfunctionofahomogeneousproblemis