Sturm-Liouville Eigenvalue Problems: Orthogonality, Eigenfunctions, and Normalization

Published on July 23, 2026

Calculus

Solve a regular Sturm-Liouville eigenvalue problem by finding eigenvalues, eigenfunctions, and normalizing them on the interval [0, pi]. Includes definitions of orthogonal and orthonormal sets.

Question

Answer

Based on the image provided, here are the step-by-step solutions for the problems presented.

Problem 1

(a) Define the following terms:
i. Orthogonal set: A set of functions

is said to be orthogonal on an interval

with respect to a weight function

if the inner product of any two distinct functions is zero:

ii. Orthonormal set: A set of functions

is orthonormal if it is an orthogonal set and the inner product of each function with itself is equal to 1:

(b) Show that is orthogonal on :
We compute the inner product for

(assuming

Since

for any integer

, the integral evaluates to

. Thus, the set is orthogonal.

© Find the orthonormal version of :
First, find the norm squared:

The norm is

. The orthonormal version is:

(d) Write the general form of a regular Sturm-Liouville operator:
The operator

is given by:

where

and

are continuous functions on

(e) State three properties of regular Sturm-Liouville eigenvalues/eigenfunctions:

The eigenvalues are real and can be ordered .
For each eigenvalue, there is a unique (up to a constant) eigenfunction.
Eigenfunctions corresponding to distinct eigenvalues are orthogonal with respect to the weight function .

(f) For , derive the eigenvalue equation:
The general solution for

(let

) is

. So .
.
Apply boundary condition at : .
Assuming , we get , which can be written as:

(g) Distinguish Dirichlet, Neumann and Robin boundary conditions:

Dirichlet: Specifies the value of the function at the boundary: .
Neumann: Specifies the value of the derivative at the boundary: .
Robin: A linear combination of the function and its derivative: .

(h) Distinguish regular and singular Sturm-Liouville problems:

Regular: are continuous on , and on the closed interval.
Singular: Occurs if the interval is infinite, or if or at one or both endpoints.

Problem 2

Given:

for

with

(a) Show that this is a Sturm-Liouville eigenvalue problem:
The equation is in the form

where

. Since

and

are positive and continuous on

, and the boundary conditions are homogeneous, it is a regular Sturm-Liouville problem.

(b) Find all eigenvalues :

Case 1: : .
. . Only trivial solution.
Case 2: : .
. . Since for , . Only trivial solution.
Case 3: : .
. .
For non-trivial solutions (), for .
Thus, eigenvalues are for .

(d) Normalize the eigenfunctions:
We need

.
Normalized eigenfunctions:

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