Section 23.4
Euler Equations
A special class of variable coefficient equations that we can solve exactly.
1
Introduction
The Cauchy-Euler equation looks like:
Notice the power of matches the order of derivative.
2
The Method
Assume x^r
Guess .
Then and .
Plugging in gives the Indicial Equation:
.
3
Three Cases
- Distinct Real Roots: .
- Repeated Root: . (Multiply by , not ).
- Complex Roots: .
.
4
Worked Examples
Example 1: Distinct Roots
Solve .
.
.
.
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Example 2: Repeated Root
Solve .
.
.
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Example 3: Complex
Solve .
.
().
.
5
Practice Quiz
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