Variation of Parameters
When the forcing function is too weird for "Guess and Check", we use the heavy artillery.
Introduction
Undetermined Coefficients is fast, but it fails for inputs like , , or .
Variation of Parameters works for any function , provided we can compute the integrals.
The Formulas
Recipe
Given , we assume the particular solution is:
The functions are found by:
Note: is the Wronskian of . Ensure the DE is in standard form ( coeff is 1) before identifying .
Visual: Green's Function Intuition
Comparison
Undetermined Coefficients
Limited Range, High Speed
Variation of Parameters
Unlimited Range, Low Speed (Hard Integrals)
We replace the constants with functions that vary over time, essentially "steering" the solution to track the forcing function.
Worked Examples
Example 1: The Classic
Solve .
1. Complementary:
. So .
2. Wronskian:
.
3. Integrals:
.
.
.
4. Particular Solution:
.
(The terms cancel).
Example 2: Repeated Roots Case
Solve . (for ).
1. Complementary:
.
2. Wronskian:
.
3. Integrals:
.
.
4. Final:
.
Example 3: Secant Forcing
Find for .
1. Basis:
. .
2. Integrals:
.
.
3. Result:
.
Practice Quiz
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