Section 22.10
Nonhomogeneous Systems
Adding external forces to the system. .
1
Introduction
Just like with single equations, the general solution is:
solves (Complementary).
is any particular solution to the full equation.
2
Method 1: Undetermined Coeffs
Guess the Vector
If , guess .
Warning: is a vector of unknown constants.
3
Method 2: Variation of Parameters
Use the Fundamental Matrix (matrix with columns as ).
This works for any , but involves inverting a matrix.
4
Worked Examples
Example 1: Undetermined Coeffs
.
1. Guess:
.
2. Plug in:
.
is wrong sign...
Correct: .
.
3. Solve:
.
.
.
.
Example 2: Fundamental Matrix
For .
Solutions are .
.
Example 3: Constant Force
, where is constant.
Ideally, steady state solution .
.
This is the "Particular Solution" (Equilibrium offset).
5
Practice Quiz
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