Homogeneous Linear SystemsExpected Educational ResultsHomogeneous Linear SystemsNormal Form of a System of Linear DEsNormal Form of a Homogenous System of Linear DEsNormal Form of a Homogenous System of Linear DEs with Constant CoefficientsSolutions to Systems of ODEsRepeated Real EigenvaluesCase 1: Repeated Real Eigenvalues with Linearly Independent EigenvectorsCase 2: Repeated Real Eigenvalues with Linearly Dependent EigenvectorsInvestigation 04CC BY-NC-SA 4.0

**Author**: John J Weber III, PhD
**Corresponding Textbook Sections**:

**Section 5.4**– Differential Operators and the Elimination Method for Systems**Section 9.4**-- Linear Systems in Nominal Form**Section 9.5**-- Homogeneous Linear Systems with Constant Coefficients**Section 9.6**-- Complex Eigenvalues

**Objective 16–1**: I can homogeneous linear systems using eigenvalues and eigenvectors.

where ,

, and

is the coefficient matrix.

**NOTE**: Lowercase variables in boldface are vectors; uppercase variables in boldface are matrices.

otherwise, the system is nonhomogenous.

This is the above equation where all elements, , of the coefficient matrix are all constants.

If is a coefficient matrix of a system of linear DEs, then is an degree polynomial in , called the **characteristic polynomial**.

**Example 02**:

Solve the following system of DEs.

**Solution**:

Rewrite system as:

**Using technology**, find the eigenvalues and corresponding eigenvectors of the coefficient matrix:x1`(* Store the matrix [3, -1, 3; 3, -1, 6; -2, 2, -2] *)`

2`A = {{3, -1, 3}, {3, -1, 6}, {-2, 2, -2}};`

3``

4`(* Display matrix *)`

5`(* To check that we entered the matrix correctly *)`

6`MatrixForm[A]`

7``

8`(* Method 1 *)`

9`(* Compute eigenvalues of matrix A *)`

10`Eigenvalues[A]`

11``

12`(* Compute eigenvectors of matrix A *)`

13`Eigenvectors[A]`

14``

15`(* Method 2 *)`

16`(* Compute eigenvalues and eigenvectors *)`

17`Eigensystem[A]`

**Warnings**:- Be very
**careful**with the*syntax*.*Syntax*is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic*Mathematica*syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf. - To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.
- The eigenvector and corresponding eigenvalues will be returned in order.

- Be very
**Using technology**, the eigenvectors of the coefficient matrix are:- For ,
- For ,
- For ,

The solution to the systems of DEs is which is equivalent to

**Example 03**:

Solve the following system of DEs:

**Solution**

Rewrite system as:

**Using technology**, the eigenvalues of the coefficient matrix are**Using technology**, the [linearly dependent] eigenvectors of the coefficient matrix are:- For ,
- For ,

The solution to the systems of DEs is which is equivalent to

Substitute the above solution into the system of DEs to solve for and .

Use matrix operations to solve the following systems of DEs:

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**Last Modified**: Monday, 19 October 2020 8:23 EDT