Section 25.2
Eigenvalues & Eigenfunctions
This is the differential equation equivalent of matrix eigenvectors.
1
Introduction
We want to solve subject to boundaries like .
The trivial solution always works.
We want to find specific values (Eigenvalues) that allow non-zero solutions (Eigenfunctions).
2
The Problem Setup
Three Possibilities
- (Oscillatory)
- (Linear)
- (Exponential)
Usually only positive works for Dirichlet conditions.
3
Worked Examples
Example 1: Dirichlet Conditions
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Try : .
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For non-trivial (), we need .
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Eigenfunctions: .
Example 2: Neumann Conditions
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Check : . Conditions give .
(Constant) works! So is an eigenvalue.
Positives: .
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Example 3: Visual Modes
Standing waves on a string of length 1.
4
Practice Quiz
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