Non-Homogeneous Equations
General Form
Structure of Solutions
Method of Undetermined Coefficients
Rules for Guessing
If (polynomial of degree ): Guess
If : Guess (unless is a root of the characteristic equation)
If or : Guess
Modification Rules
If the guess for is a solution to the homogeneous equation, multiply by (or if still a solution).
Examples
Example 1:
Homogeneous:
Guess: (since is not a root)
Example 2:
Homogeneous:
Guess: (multiply by since solves homogeneous)
Variation of Parameters
Superposition Principle
Applications
Resonance
When the forcing frequency matches the natural frequency of the system, the amplitude grows without bound (in the undamped case).