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

584 Chapter 19 Ordinary differential equations I
4 Ae 2x +Be −x −3
5 Ae −2x +Be −x + 3 2 sin2x − 1 2 cos2x,
3
2 e−2x + 3 2 sin2x − 1 2 cos2x
6 x
7 (a) Ae t +Be 5t + 3 5
(b) Ae t +Bte t + 1 2 t2 e t
8 Ae −11270t +Be −88730t
9 e −4t (Asin3t +Bcos3t)+
14sin3t +18cos3t
13
19.7 SERIESSOLUTIONOFDIFFERENTIALEQUATIONS
Wenowintroduceatechniqueforfindingsolutionsofdifferentialequationsthatcanbe
∞∑
expressed in the form of a power seriesy = a m
x m =a 0
+a 1
x +a 2
x 2 + ....Here,
a m
,m = 0,1,2,3... areconstants whosevalues needtobefound.
With the given power series form of y, we can differentiate term by term to obtain
power series for dy
dx and d2 y
dx2. By substituting these series into certain classes of differentialequationwecandeterminetheconstantsa
m
,form = 0,1,2,3...andthusobtain
the power series solution fory. This technique of representing a solution in the form of
a power series paves the way for the introduction of an important family of differential
equations,knownasBessel’sequations,whichcanbesolvedbythismethod.Thepower
series solutions so formed are known as Bessel functions which will be introduced in
Section 19.8.
Consider the following example:
Example19.30 The general solution of the differential equation d2 y
+y = 0 was shown in Example
dx2 19.16 tobe
Solution We let
y=Acosx+Bsinx
whereAandBarearbitraryconstants.Obtainthisresultbylookingforasolutionofthe
∞∑
equation inthe form ofapower seriesy = a m
x m .
y =
m=0
m=0
∞∑
a m
x m =a 0
+a 1
x+a 2
x 2 +a 3
x 3 +a 4
x 4 +a 5
x 5 +...
m=0
from which, by differentiating with respecttox,
and
∞
dy
dx = ∑
ma m
x m−1 =a 1
+2a 2
x+3a 3
x 2 +4a 4
x 3 +5a 5
x 4 +...
m=0
d 2 y
dx = ∑ ∞
(m −1)ma 2 m
x m−2 = 2a 2
+ (2)(3)a 3
x + (3)(4)a 4
x 2 + (4)(5)a 5
x 3 + ...
m=0