Section 24.4
Variation of Parameters
A powerful, general method that works even when Undetermined Coefficients fails.
1
Introduction
We seek a solution of the form:
where are found using Cramer's Rule.
2
The Formula
Using Wronskians
is the standard Wronskian.
is the Wronskian with the -th column replaced by .
3
Worked Examples
Example 1: 3rd Order Setup
.
Basis: .
.
.
Integrate .
Example 2: 2nd Order Check
Verify it matches formula.
For , puts in col 1.
.
. Correct.
Example 3: Complexity
Why not always use this?
Calculating or determinants of functions is very hard.
Integrating the result is often impossible in closed form.
Undetermined coeffs is preferred when valid.
4
Practice Quiz
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