CHAPTER 15: Boundary-Value Problems and Fourier Series
123
Thus Equation 15.12 becomes
4
n =1 np
f ( x) =
Furthermore, cos
cos 2 np ( 1) n sin npx
3
3
2p
1
4p
1
6p
= , cos
= , cos
= 1,K
3
2
3
2
3
Hence,
f ( x) =
41
px 3
2 px 2
3px
sin
sin
+ sin
L
2
3 4

122 DIFFERENTIAL EQUATIONS
c1 + c2 2 = 1
3c1e 3 + c2 e 3 = 2
whence
c1 =
3e 5
e + 3e 3
c2 =
5 + 9e 3
e + 3e 3
Finally,
y=
(3e 5)e 3 x + (5 + 9e 3 )e x
3x 2
e + 3e 3
Solved Problem 15.2 Find the eigenvalues and eigenfunctions of
y 4 ly + 4 l 2 y = 0;
y(0

120 DIFFERENTIAL EQUATIONS
Because different Sturm-Liouville problems usually generate different sets of eigenfunctions, a given piecewise smooth function will have
many expansions of the form 15.10. The basic features of all such expansions are exhibited

CHAPTER 15: Boundary-Value Problems and Fourier Series
121
For this Sturm-Liouville problem, w(x) 1, a = 0, and b = L; so that
b
L
a
0
2
w( x )e0 ( x )dx = dx = L
and
b
L
a
0
2
2
w( x )en ( x )dx = cos
npx
L
dx =
L
2
Thus 15.11 becomes
L
c0 =
1
f ( x )d

CHAPTER 15: Boundary-Value Problems and Fourier Series
117
Nontrivial solutions (i.e., solutions not identically equal to zero) to the
homogeneous boundary-value problem 15.3 exist if and only if the determinant
a1 y1 ( a) + b1 y1 ( a) 1 y2 ( a) + b1 y2 (

CHAPTER 15: Boundary-Value Problems and Fourier Series
119
Eigenfunction Expansions
A wide class of functions can be represented by innite series of eigenfunctions of a Sturm-Liouville problem.
Denition: A function f(x) is piecewise continuous on the open

118 DIFFERENTIAL EQUATIONS
a1 y( a) + b1 y ( a) = 0
a 2 y( b ) + b 2 y ( b ) = 0
(15.7)
where p(x), p(x), q(x), and w(x) are continuous on [a, b], and both p(x)
and w(x) are positive on [a, b].
Equation 15.6 can be written in standard form 15.4 by dividin

116 DIFFERENTIAL EQUATIONS
and the boundary conditions
a1 y( a) + b1 y ( a) = g1
a 2 y( b ) + b 2 y ( b ) = g 2
(15.2)
where P(x), Q(x), and f(x) are continuous in [a, b] and a1, a2, b1, b2, g1,
and g2 are all real constants. Furthermore, it is assumed th

CHAPTER 14: Numerical Methods
113
Solved Problem 14.2 Find y(1) for y = y x; y(0) = 2, using Eulers
method with h = 1 .
4
For this problem, x0 = 0, y0 = 2, and f (x, y) = y x; so Equation 14.5 be
comes yn = yn x n . Because h = 1 ,
4
x1 = x 0 + h =
1
4
x

112 DIFFERENTIAL EQUATIONS
yn +1 = f ( x n +1 , yn +1 , zn +1 )
zn +1 = g( x n +1 , yn +1 , zn +1 )
(14.18)
The derivatives associated with the predicted values are obtained similarly, by replacing y and z in Equation 14.18 with py and pz, respectively. A

110 DIFFERENTIAL EQUATIONS
Numerical Methods for Systems
First-Order Systems
Numerical methods for solving rst-order initial-value problems, including all of those described previously in this chapter, are easily extended
to a system of rst-order initial-

108 DIFFERENTIAL EQUATIONS
A predictor-corrector method is a set of two equations for yn+1. The
rst equation, called the predictor, is used to predict (obtain a rst approximation to) yn+1; the second equation, called the corrector, is then
used to obtain

CHAPTER 14: Numerical Methods
109
This is not a predictor-corrector method.
Adams-Bashforth-Moulton Method
h
(55 yn 59 yn 1 + 37 yn 2 9 yn 3 )
24
h
corrector: yn +1 = yn + (9 pyn +1 + 19 yn 5 yn 1 + yn 2 )
24
predictor: pyn +1 = yn +
(14.11)
Milnes Method

CHAPTER 14: Numerical Methods
107
Stability
The constant h in Equations 14.3 and 14.4 is called the step-size, and its
value is arbitrary. In general, the smaller the step-size, the more accurate
the approximate solution becomes at the price of more work

CHAPTER 14: Numerical Methods
105
Direction Fields
Graphical methods produce plots of solutions to rst-order differential
equations of the form
y = f ( x, y)
(14.1)
where the derivative appears only on the left side of the equation.
Example 14.1 (a) For t

106 DIFFERENTIAL EQUATIONS
Eulers Method
If an initial condition of the form
y(x0) = y0
(14.2)
is also specied, then the only solution curve of Equation 14.1 of interest is the one that passes through the initial point (x0, y0).
To obtain a graphical appr