Section 24.3
Undetermined Coefficients
If the forcing function is a polynomial, exponential, or sine/cosine, we can guess the form of the solution.
1
Introduction
The logic is identical to 2nd order equations.
should "look like" and its derivatives.
2
The Rules
The Multiplier Rule
If your guess duplicates a term in the complementary solution , multiply by .
is the smallest integer that removes the duplication.
3
Worked Examples
Example 1: Basic Guess
.
1. : . .
2. Guess: is a polynomial (deg 1).
Base guess .
Conflict: overlaps with (constant).
New guess: .
Example 2: Resonant Frequency
.
has terms (from ).
Forcing matches .
Guess must be multiplied by x: .
Example 3: Superposition
.
Solve for separately ().
Solve for separately ().
Add them up.
4
Practice Quiz
Loading...