2008_Fourier Series(2)
Naval Architecture & Ocean Enginee ring
Engineering Mathematics 2
Prof. Kyu-Yeul Lee
Department of Naval Architecture and Ocean Engineering, Seoul National University of College of Engineering
[2008][10-1]
November, 2008
2008_Fourier Series(2)
Naval Architecture & Ocean Enginee ring
Fourier Series(2)
: Sturm-Liouville Problem
2008_Fourier Series(2)
Sturm-Liouville Problem
Review
0
General solutions Linear Equations
0
y y
2
0
y y
2
0
y
y 0
e
xc y
1 x c
x c
y
1cos
2sin
) sinh
cosh
,
2 1
2 1
x c
x c
y
or e
c e
c
y
x x
General solutions Cauchy-Euler Equation
2
0
2
y x y y
x
0 ,
ln
0 ,
2 1
2 1
x c
c y
x c x
c 0 y
0 x
Linear Equations
When is a finite interval
x
When is an infinite or half finite interval
x
2008_Fourier Series(2)
Sturm-Liouville Problem
Review
General solutions Parametric Bessel equation
) ( )
(
2 00
1
J x c Y x
c
y
Particular solutions are polynomials
Legendre‟s equation
0 )
1 (
2 )
1
( x
2y x y n n y
0 1
2 2
( ) 1, ( ) ,
( ) 1 (3 1), 2
y P x
y P x x
y P x x
0 x
0
2 2 2
0 x y y x y
,...
2 , 1 ,
0
n
2008_Fourier Series(2)
Sturm-Liouville Problem
Eigenvalues and Eigenfunctions
0 )
( ,
0 )
0 ( ,
0
" y y y L y
When (Case III ) 0 0
2 ,
Write
x c
x c
y 1 cos 2 sin
or
1 0 c
2 sin
n
y c n x
L
(nontrivial solution)
m i i
m
1 ,
2
Then roots of auxiliary equation is
2 0 c 0
) 0 ( y
2 2
) (
, L
n n
L n n
0 )
( L y
Recall example 2 of section 3.9
Eigenvalues
Eigenfunctions
2008_Fourier Series(2)
Sturm-Liouville Problem
Eigenvalues and Eigenfunctions
0, (0) 0, ( ) 0 y y y y L
2 sin
n
y c n x
L
2 2
) ( L
n
n n
Eigenvalues
Eigenfunctions
It is important to recognize the set of functions generated by this B.V.P the orthogonal set of functions on the interval used as the basis for the Fourier sine series
) , 0 ( L
0, (0) 0, ( ) 0
y y y y L the Fourier cosine series
2008_Fourier Series(2)
Sturm-Liouville Problem
Example 1
Eigenvalues and Eigenfunctions
0 ) ( , 0 ) 0 ( ,
0
y y y L
y
It is left as an exercise to show, by considering the three possible
cases for the parameter (zero, negative, or positive; that is,
that the eigenvalues and )
eigenfunctions for the boundary- value problem
0 ,
0 ,
0 ,
0 ,
0 2 2
and
are, respectively,
is an
eigenvalue for this BVP and is the corresponding eigenfunction.
The latter comes from solving subject to the same boundary
conditions . Note also that can be incorporated into the family by
permitting . The set
is orthogonal on the interval [0,L].
, , 2 , 1 , 0 ,
/
22 2 2
nn L n
n
1
cos( / ) ,
10.
00
y c n x L c
1 y
0 y
0 ) ( , 0 ) 0
(
y L
y
1 y
) / cos( n x L y
0
n {cos( n x / L )}, ,
, 3 , 2 , 1 ,
0
n
2008_Fourier Series(2)
Sturm-Liouville Problem
Regular Sturm-Liouville Problem p , q , r , r
real-valued functions
continuous on an interval
are not both zero
] , [ b a
1 1
, B A Solve : [ r ( x ) y ] [ q ( x ) p ( x )] y 0
dx
d
Subject to:
0 )
( )
(
0 )
( )
(
2 2
1 1
b y B b
y A
a y B a
y A
0 ) ( , 0 )
( x p x r
for every in the interval x [ b a , ]
are not both zero
2 2
, B A
0, (0) 0, ( ) 0 y y y y L
L b
a B
A B
A
1 1 ,
1 0 ,
2 1 ,
2 0 , 0 ,
L b
a B
A B
A
1 0 ,
1 1 ,
2 0 ,
2 1 , 0 ,
0, (0) 0, ( ) 0 y y y y L
1 )
( , 0 )
( , 1 )
( x q x r x p
Special case
B.V.P
Sturm-Liouville Problem :
Homogeneous Boundary Value problem -> Trivial solution y=0
-> goal : find nontrivial solution y
(Eigenvalues, Eigenfunctions)
2008_Fourier Series(2)
Sturm-Liouville Problem
“Homogeneous”
Homogeneous D.E.+
Homogeneous B/C
nonzero C
C a
y B a
y
A Nonhomogeneous B/C
1( )
1 ( )
2,
2:
Regular Sturm-Liouville Problem p , q , r , r
real-valued functions
continuous on an interval
are not both zero
] , [ b a
1 1
, B A Solve : [ r ( x ) y ] [ q ( x ) p ( x )] y 0
dx
d
Subject to:
0 )
( )
(
0 )
( )
(
2 2
1 1
b y B b
y A
a y B a
y A
0 ) ( , 0 )
( x p x r
for every in the interval x [ b a , ]
are not both zero
2 2
, B
A
B.V.P
2008_Fourier Series(2)
12.5 Sturm-Liouville Problem
“trivial solution is not our interest”
Homogeneous B.V.P always
possesses the trivial solution y 0
Regular Sturm-Liouville Problem p , q , r , r
real-valued functions
continuous on an interval
are not both zero
] , [ b a
1 1
, B A Solve : [ r ( x ) y ] [ q ( x ) p ( x )] y 0
dx
d
Subject to:
0 )
( )
(
0 )
( )
(
2 2
1 1
b y B b
y A
a y B a
y A
0 ) ( , 0 )
( x p x r
for every in the interval x [ b a , ]
are not both zero
2 2
, B
A
B.V.P
2008_Fourier Series(2)
Sturm-Liouville Problem
By utilizing the inner product
Properties of the Regular Sturm-Liouville Problem
(a) There exist an infinite number of real eigenvalues that can be arranged in increasing
order such that as (b) For each eigenvalues there is only one eigenfunction (except for nonzero constant
multiples)
(c) Eigenfunctions corresponding to different eigenvalues are linearly independent
(d) The set of eigenfunctions corresponding to the set of eigenvalues is orthogonal with respect to the weight function on interval
Theorem 12.3
) (x p
n
1 2 3
n n
] , [ b a
Solve : [ r ( x ) y ] [ q ( x ) p ( x )] y 0 dx
d
Subject to:
0 )
( )
(
0 )
( )
(
2 2
1 1
b y B b
y A
a y B a
y
A
2008_Fourier Series(2)
Sturm-Liouville Problem
n b
a
p x y
mx y
nx dx
m
( ) ( ) ( ) 0 ,
Proof of (d)
) 1 ( 0 )]
( )
( [ ] )
(
[ r x y
m q x
mp x y
m dx
d
) 2 ( 0 )]
( )
( [ ] ) (
[ r x y
n q x
np x y
n dx
d
: )
2 ( )
1
( y
n y
m nd [ ( )
m]
md [ ( ) ] (
n m n) ( )
n m0
y r x y y r x y p x y y
dx dx
(
n m) ( )
n m nd [ ( )
m]
md [ ( ) ]
np x y y y r x y y r x y
dx dx
[ ( ) ] [ ( ) ] [ ( ) ] [ ( ) ]
n m m n m n n m
d d d d
y r x y r x y y y r x y r x y y
dx dx dx dx
n[ ( )
m]
m[ ( ) ]
n
d d
y r x y y r x y
dx dx
( )
n mr x y y r x y y ( )
n
m(d) The set of eigenfunctions corresponding to the set of eigenvalues is orthogonal with respect to the weight function p(x) on interval[ ba, ]
2008_Fourier Series(2)
Sturm-Liouville Problem
n b
a
p x y
mx y
nx dx
m
( ) ( ) ( ) 0 ,
Integrating
ba n m m n
b
a n m
m
n
y r x y dx
dx y d
x r dx y
dx d y y x
p ( ) [ ( ) ] [ ( ) ]
) (
( )[ ( ) ( )] ( )[ ( ) ( )]
( )[ ( ) ( )] ( )[ ( ) ( )]
n m n m
m n m n
y b r b y b y a r a y a y b r b y b y a r a y a
)]
( ) ( )
( ) ( )[
(
)]
( ) ( )
( ) ( )[
(
a y a y a
y a y a r
b y b y b
y b y b r
n m
n m
n m
n m
0 )
( )
(
0 )
( )
(
2 2
1 1
b y B b
y A
a y B a
y A
Boundary Condition
) 4 ( 0 )
( )
(
) 3 ( 0 )
( )
(
1 1
1 1
a y B a
y A
a y
B a
y A
n n
m m
) 6 ( 0 )
( )
(
) 5 ( 0
) ( )
(
2 2
2 2
b y B b
y A
b y B b
y A
m n
m m
Proof of (d)
(
n m) ( )
n md
n[ ( )
m] d
m[ ( ) ]
np x y y y r x y y r x y
dx dx
(d) The set of eigenfunctions corresponding to the set of eigenvalues is orthogonal with respect to the weight function p(x) on interval[ ba, ]
2008_Fourier Series(2)
Sturm-Liouville Problem
n b
a
p x y
mx y
nx dx
m
( ) ( ) ( ) 0 ,
dx y
y x
b
p
a n m
m
n
) ( )
(
Boundary Condition
As are not both zero A 1 , B 1
0 )
( ) ( )
( )
( a y a y a y a
y
m n m n) 4 ( 0
) ( )
(
) 3 ( 0
) ( )
(
1 1
1 1
a y B a
y A
a y
B a
y A
n n
m
m
0 0 )
( )
(
) ( )
(
1 1
B A a
y a
y
a y
a y
n n
m m
) ( )
(
) ( )
det (
a y a
y
a y
a y
n n
m m
)]
( ) ( )
( ) ( )[
(
)]
( ) ( )
( ) ( )[
(
a y a y
a y a y
a r
b y b y b
y b y b r
n m
n m
n m
n m
Proof of (d)
(d) The set of eigenfunctions corresponding to the set of eigenvalues is orthogonal with respect to the weight function p(x) on interval[ ba, ]
2008_Fourier Series(2)
Sturm-Liouville Problem
n b
a
p x y
mx y
nx dx
m
( ) ( ) ( ) 0 ,
dx y
y x
b
p
a n m
m
n
) ( )
(
Boundary Condition
As are not both zero A 2 , B 2
0 )
( ) ( )
( )
( b y b y b y b
y
m n m n
0 0 )
( )
(
) ( )
(
2 2
B A b
y b
y
b y b
y
n n
m m
) ( )
(
) ( )
det (
b y b
y
b y b
y
n n
m m
) 6 ( 0
) ( )
(
) 5 ( 0
) ( )
(
2 2
2 2
b y B b
y A
b y B b
y A
n n
m m
)]
( ) ( )
( ) ( )[
(
)]
( ) ( )
( ) ( )[
(
a y a y
a y a y
a r
b y b y b
y b y b r
n m
n m
n m
n m
Proof of (d)
(d) The set of eigenfunctions corresponding to the set of eigenvalues is orthogonal with respect to the weight function p(x) on interval[ ba, ]
2008_Fourier Series(2)
Sturm-Liouville Problem
n b
a
p x y
mx y
nx dx
m
( ) ( ) ( ) 0 ,
dx y
y x
b
p
a n m
m
n
) ( )
(
From Boundary Condition:
0 )
( ) ( )
( )
( b y b y b y b
y
m n m n)]
( ) ( )
( ) ( )[
(
)]
( ) ( )
( ) ( )[
(
a y a y
a y a y
a r
b y b y b
y b y b r
n m
n m
n m
n m
0 )
( ) ( )
( )
( a y a y a y a
y
m n m nzero zero
0 )
( )
(
n
m
abp x y
ny
mdx
m n
b
a
p x y
ny
mdx
( ) 0 ,
Proof of (d)
(d) The set of eigenfunctions corresponding to the set of eigenvalues is orthogonal with respect to the weight function p(x) on interval[ ba, ]
2008_Fourier Series(2)
Sturm-Liouville Problem
n b
a
p x y
mx y
nx dx
m
( ) ( ) ( ) 0 ,
dx y
y x
b
p
a n m
m
n
) ( )
(
From Boundary Condition:
0 )
( ) ( )
( )
( b y b y b y b
y
m n m n)]
( ) ( )
( ) ( )[
(
)]
( ) ( )
( ) ( )[
(
a y a y
a y a y
a r
b y b y b
y b y b r
n m
n m
n m
n m
0 )
( ) ( )
( )
( a y a y a y a
y
m n m nzero zero
0 )
( )
(
n
m
abp x y
ny
mdx
m n
b
a
p x y
ny
mdx
( ) 0 ,
Orthogonal relation Proof of (d)
(d) The set of eigenfunctions corresponding to the set of eigenvalues is orthogonal with respect to the weight function p(x) on interval[ ba, ]
2008_Fourier Series(2)
Sturm-Liouville Problem
Example 2
A Regular Sturm-Liouville Problem
Slove the boundary-value problem
. 0 ) 1 ( )
1 ( , 0 ) 0 ( ,
0
y y y y
y
, 0 ,
0
0
2
and where
0 y solution trivial
the
, 0 ,
2
0
x c
x c
y is
y y
of solution general
the
sin
cos
0
2 1
2
2
sin
2cos
2(sin cos ) 0 c c c
2
0 c
The second boundary condition is satisfied if
0 ) 1 ( )
1
( y y
Choosing , we see that the
last equation is equivalent to tan
The eigenvalues of problem are
then , where , , are the consecutive positive roots of
2 n
n
nn 1 , 2 , 3 ,
, ,
,
2 31
tan
1 2
(0) 0 sin
y c y c x
y
tan
y
2008_Fourier Series(2)
Sturm-Liouville Problem
Example 2
A Regular Sturm-Liouville Problem
Slove the boundary-value problem
. 0 ) 1 ( )
1 ( , 0 ) 0 ( ,
0
y y y y
y
, 0 ,
0
0
2
and where
0 y solution trivial
the
, 0 ,
2
0
x c
x c
y is
y y
of solution general
the
sin
cos
0
2 1
2
2
sin
2cos
2(sin cos ) 0 c c c
2
0 c
The second boundary condition is satisfied if
0 ) 1 ( )
1
( y y
Choosing , we see that the
last equation is equivalent to tan
and the corresponding solutions are
1 2 . 0288 ,
2 4 . 9132 ,
3 7 . 9787 ,
4 11 . 0855 ,
, 9787 .
7 sin ,
9132 .
4 sin ,
0288 .
2
sin
2 31
x y x y x
y
x y
4 sin 11 . 0855
In general, the eigenfunctions of the problem are {sin
n}, n 1 , 2 , 3 ,
1 2
(0) 0 sin
y c y c x
y
tan
y
2008_Fourier Series(2)
Sturm-Liouville Problem
Example 2
A Regular Sturm-Liouville Problem
Slove the boundary-value problem
. 0 ) 1 ( )
1 ( , 0 ) 0 ( ,
0
y y y y
y
1 1 2 2
( ) 1, ( ) 0, ( ) 1
1, 0, 1, 1
r x q x p x
A B A B
Regular Sturm-Liouville Problem
Solve :
[ r ( x ) y ] [ q ( x ) p ( x )] y 0 dx
d
Subject to:
0 ) ( )
(
0 ) ( )
(
2 2
1 1
b y B b y A
a y B a y A
B.V.P
m n
b
a
p x y
ny
mdx
( ) 0 ,
Orthogonal relation Regular Sturm-Liouville Problem
In general, the eigenfunctions of the problem are {sin
n}, n 1 , 2 , 3 ,
, 3 , 2 , 1 },
{sin
nn
is an orthogonal set with respect to the weight functionon the interval [0,1].
( ) 1
p x
2008_Fourier Series(2)
Sturm-Liouville Problem
dx y
y x
b
p
a n m
m
n
) ( )
(
should be zero for Orthogonal relation
)]
( ) ( )
( ) ( )[
(
)]
( ) ( )
( ) ( )[
(
a y a y
a y a y
a r
b y b y b
y b y b r
n m
n m
n m
n m
m n
b
a
p x y
ny
mdx
( ) 0 ,
Orthogonal relation
0 )]
( )
( [ ] ) (
[ r x y q x p x y dx
d ( ( ) ) ( ( ) ) 0 0
2 2
1 1
b y B b
y A
a y B a
y A
Boundary Condition:
Regular Sturm-Liouville Problem
In some circumstances, we can prove the orthogonality of the solutions of without the necessity of specifying a
boundary condition at x=a and at x=b
0 )]
( )
( [ ] ) (
[ r x y q x p x y dx
d
Singular Sturm-Liouville Problem
]
,
[ b a
2008_Fourier Series(2)
Sturm-Liouville Problem
dx y
y x
b
p
a n m
m
n
) ( )
(
should be zero for Orthogonal relation
)]
( ) ( )
( ) ( )[
(
)]
( ) ( )
( ) ( )[
(
a y a y
a y a y
a r
b y b y b
y b y b r
n m
n m
n m
n m
m n
b
a
p x y
ny
mdx
( ) 0 ,
Orthogonal relation
0 )]
( )
( [ ] ) (
[ r x y q x p x y dx
d
If then may be a singular and the equation
may become unbounded as
however
0 )
( a r
0 )
( )
(
0 )
( )
(
2 2
1 1
b y B b
y A
a y B a
y A
)]
( ) ( )
( ) ( )[
( )]
( ) ( )
( ) ( )[
( b y b y b y b y b r a y a y a y a y a
r
m
n
m n
m
n
m n
dropped from the problem : no boundary condition at
Singular Sturm-Liouville Problem
Orthogonal relation hold on
0 )]
( )
( [ ] ) (
[ r x y q x p x y dx
d
a x
a x
] ,
[ b a x a
] , (
, a b
Boundary Condition:
Regular Sturm-Liouville Problem
zero
]
,
[ b a
2008_Fourier Series(2)
Sturm-Liouville Problem
If then may be a singular and the equation
may become unbounded as
however
0 )
( b r
Orthogonal relation hold on
0 )]
( )
( [ ] ) (
[ r x y q x p x y dx
d
b x
b x
] , [ b a
)]
( ) ( )
( ) ( )[
( )]
( ) ( )
( ) ( )[
( b y b y b y b y b r a y a y a y a y a
r
zerom
n
m n
m
n
m n
dropped from the problem
: no boundary condition at
x b
) , [
, a b
Boundary Condition:
dx y
y x
b
p
a n m
m
n
) ( )
(
should be zero for Orthogonal relation
)]
( ) ( )
( ) ( )[
(
)]
( ) ( )
( ) ( )[
(
a y a y
a y a y
a r
b y b y b
y b y b r
n m
n m
n m
n m
m n
b
a
p x y
ny
mdx
( ) 0 ,
Orthogonal relation
0 )]
( )
( [ ] ) (
[ r x y q x p x y dx
d ( ( ) ) ( ( ) ) 0 0
2 2
1 1
b y B b
y A
a y B a
y A
Singular Sturm-Liouville Problem
Regular Sturm-Liouville Problem
]
,
[ b a
2008_Fourier Series(2)
Sturm-Liouville Problem
m n
b
a
p x y
ny
mdx
( ) 0 ,
Orthogonal relation
0 )]
( )
( [ ] ) (
[ r x y q x p x y dx
d
example*)
Singular Sturm-Liouville Problem
Legendre‟s equation is a Sturm-Liouville equation
( 1 x
2) y y 0 ( 1 x
2) y 2 x y y 0
) (x r 0
) 1 ( r
Since need no boundary conditions, but have a singular Sturm-Liouville problem on the interval . We know that , the Legendre polynomials are solutions of the problem for
Hence these are the eigenfunctions. They are orthogonal on the interval
1 1
x
) 1 (
n n
) (x P
n,...)
3 2 , 2 1 , 0 ,...(
2 , 1 ,
0
n
0 )
1 ( 2
) 1
( x
2y x y n n y Legendre‟s equation
) (
, 0 )
( )
1
(
1
p
mx p
nx dx m n
]
,
[ b a
2008_Fourier Series(2)
Sturm-Liouville Problem
If then
Periodic r ( a ) Sturm-Liouville Problem r ( b )
∴Orthogonal relation hold on with
[ b a , ]
Boundary Condition:
( ( ) ( ) ( ) ( )) ( ( ) ( ) ( ) ( ))
) (
)]
( ) ( )
( ) ( )[
( )]
( ) ( )
( ) ( )[
(
b y b y a
y a y
a y a y
b y b y a
r
a y a y
a y a y
a r b
y b y b
y b y b r
n m
n m
n m
n m
n m
n m
n m
n m
) ( )
( ),
( )
( a y b y a y b
y
dx y
y x
b
p
a n m
m
n
) ( )
(
should be zero for Orthogonal relation
)]
( ) ( )
( ) ( )[
(
)]
( ) ( )
( ) ( )[
(
a y a y
a y a y
a r
b y b y b
y b y b r
n m
n m
n m
n m
m n
b
a
p x y
ny
mdx
( ) 0 ,
Orthogonal relation
0 )]
( )
( [ ] ) (
[ r x y q x p x y dx
d ( ( ) ) ( ( ) ) 0 0
2 2
1 1
b y B b
y A
a y B a
y
Regular Sturm-Liouville Problem A
]
,
[ b a
2008_Fourier Series(2)
Sturm-Liouville Problem
with
with boundary Condition:
( ) ( ), ( ) ( ) y a y b y a y b
dx y y x
b
p
a n m
m
n
) ( )
(
)]
( ) ( )
( ) ( )[
(
)]
( ) ( )
( ) ( )[
(
a y a y a
y a y a r
b y b y b
y b y b r
n m
n m
n m
n m
m n
b
a
p x y
ny
mdx
( ) 0 ,
Orthogonal relation hold on.. [a,b]
0 )]
( )
( [ ] ) (
[ r x y q x p x y dx
d
0 ) ( )
(
0 ) ( )
(
2 2
1 1
b y B b
y A
a y B a
y A
Sturm-Liouville Problem
• Regular
• Singular
• Periodic
0 )
( a r
0 )
( b r
] , [ b a
) ( )
( a r b r
( ) 0 r x
] , [ b a
without B/C at x=a without B/C at x=b
2
( )
2( ) 0
A y b B y b
1
( )
1( ) 0
A y a B y a
By assuming the solution (y) are bounded on the closed interval [a,b] , then
2008_Fourier Series(2)
Sturm-Liouville Problem
Self-Adjoint Form
If the coefficient are continuous and for all in some interval, then any second-order differential equation
can be recast into the so-called „self-adjoint form‟.
0 )
( x a
[ ( ) ] [ ( ) ( ) ] 0 d r x y q x p x y
dx
0 ))
( )
( ( )
( )
( x y b x y c x d x y
a
x
Recall, ch. 2.3 integrating factor
1
( )
0( ) 0
a x y a x y [ y ] 0 dx
d
) (
) ) (
( ,
1 ) 0
(
x a
x x a
p
e
p x dx
2008_Fourier Series(2)
Sturm-Liouville Problem
Self-Adjoint Form
0 )]
( )
( [ ] ) (
[ r x y q x p x y dx
d
0 ))
( )
( ( )
( )
( x y b x y c x d x y
a
dx
x a
x b
e
( )) (
) 0 (
) ( )
( ) ( )
( )
(
y
x a
x d x
a x y c
x a
x
y b
) 0 (
) ( )
( ) ( )
( )
(
(( )) (( )) (( ))) (
) (
y
x a
x e d
x a
x e c
y x e
a x y b
e
x dx a
x dx b
x a
x dx b
x a
x dx b
x a
x b
) 0 (
) ( )
( )
(
(( )) (( ))) (
) (
e
y
x a
x e d
x a
x y c
dt e
d
ab xx dx ab xx dx
ba xx dxmultiply
) (x
r q (x ) p x ( )
( ) a x
divided by
2008_Fourier Series(2)
Sturm-Liouville Problem
Self-Adjoint Form
0 )]
( )
( [ ] ) (
[ r x y q x p x y dx
d
0 ))
( )
( ( )
( )
( x y b x y c x d x y
a
a x dx x bx e a
x x d
p
( )) (
) (
) ) (
(
a x dx x bx e a
x x c
q
( )) (
) (
) ) (
(
a x dx x be x
r
( )) (
)
Example)
3 y 6 y y 0 (
x dx dx
x a
x b
e e
e
( ) 2 2) (
3 0 3
6
y y
y
3 0
2
2 22
y e y e y
e
x x
x2
2
3 0
x
d
xe y e
dx y
2008_Fourier Series(2)
Sturm-Liouville Problem
1
ln
0
dx x
e
xe x
x
General solution
y c
1J
n( x ) c
2Y
n( x )
Ex.)Parametric Bessel Series*
,...
2 , 1 , 0 ,
0 )
(
2 2 22
y x y x n y n
x
converges on when converges on
) (x
J
n[ 0 , ) )
(x
Y
n( 0 , )
0 n
0 )
1 (
2 2
2
y
x y n
y x
multiply
0 )
(
2
2
y
x x n
y y
x
) (x
r q (x ) p (x )
(
2
2x ) y 0
x y n
dx x
d
Self-Adjoint Form
x
2divided by
m n b
a
p x y
ny
mdx
( ) 0 ,
0 )]
( ) ( [ ] ) (
[r x y q x p x y dx
d