# PERTEMUAN 8 STRUM-LIOVILLE SYSTEMS Matakuliah : K0124 / Matematika Teknik II

```Matakuliah
Tahun
: K0124 / Matematika Teknik II
: 2006/2007
PERTEMUAN 8
STRUM-LIOVILLE SYSTEMS
1
A boundry value problem having the form
d 
dy 
p
(
x
)
 q ( x)  r ( x)y  0


dx 
dx 
axb



a1 y (a )  a 2 y ' (a)  0, b1 y (b)  b2 y ' (b)  0

2
where a1 , a2 , b1 , b2 are given constants; p( x), q( x), r ( x)
are given functions which we shall assume to be
differentiable and

is an unspecified parameter
independent of x, is called a Sturm-Liouville
boundry-value problem or Sturm-Liouville System.
3
A non-trivial solution of this system, i.e. one which is
not identically zero, exist in general only for a
particular set of values of the parameter . These
values are called the characteristic values, or more
often eigenvalues, of the system.
4
The corresponding solution are called characteristic
functions or eigenfunctions of the system. In general
to each eigenvalue there is aone eigenfunction,
although exceptions can occur.
5
If p(x), q(x) are real, then the eigenvalues are real.
Also the eigenfunctions form an orthogonal set with
respect to the density function r(x) which is generally
taken as non-negative , i.e. r(x)  0.
6
It follows that by suitable normalization the set of
functions can be made an orthonormal set with
respect to r(x) in axb .
7
TERIMA KASIH
8
```