Section 22.5
Solutions to Systems
Just like solves , we will find that solves .
1
Introduction
We guess a solution of the form:
Plugging this into gives , or .
This is why eigenvalues matter!
2
General Solution
Superposition
If is and has linearly independent eigenvectors , the general solution is:
3
Visual: Phase Portrait
Interactive: Flow
The solution curve is a combination of flow along the "Eigen-axes".
4
Worked Examples
Example 1: Solving Diagonal System
Solve .
1. Decoupled:
.
.
2. Vector Form:
.
Eigenvalues are 3, -2. Eigenvectors are standard basis.
Example 2: General Solution
Given eigenvalues and . Write general solution.
1. Assemble:
.
.
.
Example 3: IVP
Use Example 2 with .
1. Plug in t=0:
.
2. Solve System:
.
.
Subtract: . Then .
3. Final Answer:
.
5
Practice Quiz
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