A Sturm-Liouville Eigenvalue Problem

c

2.6

n

2.8

This Demonstration shows solutions to the Sturm–Liouville eigenvalue problem

ℒy=(xy')'+

2

x

y=−λσy

subject to the boundary conditions

y'(1)=0

and

y'(2)=0

.

Details

This equation may be written as a Cauchy–Euler equation

2

x

y''+xy'+(λ+2)y=0

, where

λ+2>0

.

Solutions are of the form

y(x)=ccos

2πn

log2

log(x)

.

You can vary the parameters

c

and

n

.

The eigenvalues are

λ=

4

2

n.

2

π

2

(log2)

-2

.

References

[1] M. Al-Gwaiz, Sturm–Liouville Theory and Its Applications, London: Springer, January 2008.

[2] A. Zettl, Sturm–Liouville Theory, Providence, RI: American Mathematical Society, 2005.

[3] M. L. Abell and J. P. Braselton, Differential Equations with Mathematica, 3rd ed., Boston: Elsevier, 2004.

[4] R. L. Herman, A Second Course in Ordinary Differential Equations: Dynamical Systems and Boundary Value Problems, Monograph, December 2008. (Oct 1, 2020) people.uncw.edu/hermanr/mat463/ODEBook/Book/ODE_Main.pdf.

External Links

Solve a Basic Sturm–Liouville Problem

Permanent Citation

Tania Mata, Rafael Fernandez, Tomas Garza, Rafael Fernandez

"A Sturm-Liouville Eigenvalue Problem"
http://demonstrations.wolfram.com/ASturmLiouvilleEigenvalueProblem/
Wolfram Demonstrations Project
Published: October 12, 2020