Nonhomogeneous Equations
When external forces act on a system, the solution is a mix of the system's natural behavior and its response to the force.
Introduction
We are now solving equations where :
The function represents an external forcing function (like gravity, a motor, or a voltage source).
Structure of the Solution
Theorem
The general solution to a nonhomogeneous equation is:
- (Complementary Solution): Solves the homogeneous version (). Contains constants . Represents natural recurrence.
- (Particular Solution): Any single function that solves the full equation. Contains no arbitrary constants. Represents the forced response.
Visual: Natural + Forced
Interactive: Superposition
The "Complementary" part handles initial conditions (transient). The "Particular" part tracks the input forcing (steady-state).
Worked Examples
Example 1: Verifying Decomposition
For :
1. Find .
2. Verify works.
3. Write general solution.
1. Complementary Solution:
Solve .
.
.
2. Check Particular:
.
.
Matches RHS. Verified.
3. General Solution:
.
Example 2: IVP Breakdown
Solve with .
1. Find :
(Double root at 0).
2. Find :
By inspection, gives .
3. General Solution:
.
4. Apply ICs:
.
.
.
So .
Example 3: Superposition of Forces
If solves and solves , what solves ?
1. Principle:
Since the operator is Linear...
.
2. Answer:
.
This allows us to break complex forcing functions into simpler pieces.
Practice Quiz
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