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

586 Chapter 19 Ordinary differential equations I
This method of solution iswidely applicable and involves the following steps:
• given a differential equation, we seek a solution in the form of the power series
∞∑
y = a m
x m ,
m=0
• thispower series issubstitutedinto the differential equation
• the coefficientsa m
are then found by the technique of equating coefficients.
Once the values of a m
are found, the solution follows in the form of a power series.
Some differential equations and their solutions have such importance in engineering
and scientific applications that the series obtained are used to introduce new functions
in much the same way as the sine and cosine functions appeared in Equation 19.24.
We shall see in the following section how an important class of functions known as
Besselfunctionsarisesinthisway.Thereisofcoursethequestionofwhetheranypower
series expansion converges but such discussion is beyond the scope of the introductory
treatment given here.
EXERCISES19.7
1 Thegeneral solution ofthe equation d2 y
dx 2 −y=0is
y =Ae x +Be −x whereAandBarearbitraryconstants.
Obtain thisresultby lookingforasolution ofthe
∞∑
equation in the form ofapower seriesy = a m x m .
2 Thegeneral solution ofthe equation
m=0
d 2 y
dx 2 −3dy dx +2y = 0 isy =Aex +Be 2x whereAand
Barearbitraryconstants.Obtainthisresultbylooking
forasolution ofthe equation in the formofapower
∞∑
seriesy = a m x m .
m=0
3 Verifyusingthemethodofseparationofvariablesthat
the generalsolutionof dy
dx = 2xyisy =Kex2 whereK
isan arbitraryconstant. Obtainthe same resultby
seeking apower seriessolution ofthe differential
∞∑
equationin the formy = a m x m .
m=0
4 Obtainthe first fournon-zeroterms in the power
seriessolution ofthe initialvalue problem
dy
dx +xy=0,y(0)=1.
5 Obtainthe power series solution ofthe equation
d 2 y
dx 2 +xy = 0,upto termsinvolvingx7 . This
differentialequation isknown asanAiryequation.
Solutions
4 y=1− x2
2 + x4
8 − x6
48 +.... 5
y=a 0
(
1 − x3
6 + x6
180 −... )
(
)
+a 1 x − x4
12 + x7
504 −... .