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Sturm-Liouville Eigenvalue Problem: Solving Boundary Value ODEs
Published on July 23, 2026
Calculus
Solve the Sturm-Liouville eigenvalue problem y'' + lambda y = 0 with Dirichlet boundary conditions, finding eigenvalues and eigenfunctions.
Question
Answer
Based on the image provided, here are the step-by-step solutions for the problems presented in section 2.
Problem 2
Consider the boundary value problem:
with boundary conditions:
(a) Show that this is a Sturm-Liouville eigenvalue problem.
A regular Sturm-Liouville problem is defined by a differential equation of the form:
on an interval
Comparison:
Our equation is
Our equation is
- Comparing terms, we have
, , and . - Since
and for all , and all functions are continuous, it fits the form. - The boundary conditions
and are standard separated homogeneous Dirichlet boundary conditions. - Therefore, this is a regular Sturm-Liouville eigenvalue problem.
(b) Find all eigenvalues
We consider the three cases suggested:
Case 1:
The equation becomes
The general solution is
The equation becomes
The general solution is
- Applying
. - Applying
.
Since, we only have the trivial solution . Thus, is not an eigenvalue.
Case 2:
The equation becomes
The general solution is
The equation becomes
The general solution is
- Applying
. - Applying
.
Sinceand , . Thus, must be .
We only have the trivial solution, soyields no eigenvalues.
Case 3:
The equation becomes
The general solution is
The equation becomes
The general solution is
- Applying
. - Applying
.
For a non-trivial solution, we need, which implies .
This occurs whenfor
So,.
Since, the eigenvalues are:
© Find the corresponding eigenfunctions.
From Case 3, we found
The eigenfunctions are:
The eigenfunctions are:
(Usually, we represent the basis eigenfunctions by setting the arbitrary constant
(d) Normalize the eigenfunctions.
To normalize, we find
Using the identity
The normalized eigenfunctions are: