Undetermined Coefficients
If the forcing function is nice (exponential, polynomial, or sine/cosine), we can guess the form of the solution and solve for the details.
Introduction
This method works for constant coefficient equations where is of a special form.
The key idea: "Output resembles Input".
If we force a system with , the response probably looks like .
The Guessing Rules
| Forcing Function | Guess for |
|---|---|
| or | |
Reviewing Exception:
If your guess is already part of (the Complementary Solution), multiply your guess by (or ) until it is unique.
Visual: Fitting the Force
Interactive: Finding the Amplitude
We guess the shape (exponential), then simply tune the amplitude (A) until the equation balances.
Worked Examples
Example 1: Basic Exponential
Solve .
1. Complementary Solution:
.
.
2. Particular Guess:
Guess . (Not in , so safe).
, .
3. Plug In:
.
.
.
4. Final:
.
Example 2: Trig Forcing
Solve .
1. Complementary:
. .
2. Guess:
Guess .
.
.
3. Plug In:
.
.
4. Match Coefficients:
.
.
.
Example 3: The Glitch
Solve .
1. Complementary:
. .
2. Initial Guess:
Try . This is already in ! It will give 0.
3. Correct Guess:
Multiply by . .
.
.
4. Plug In:
.
The terms cancel (magic!).
.
.
Practice Quiz
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