Homogeneous Linear Systems

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 ODEsDistinct Real EigenvaluesInvestigation 01CC BY-NC-SA 4.0

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

Expected Educational Results

Homogeneous Linear Systems

Normal Form of a System of Linear DEs


where ,

, and

is the coefficient matrix.

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

Normal Form of a Homogenous System of Linear DEs

otherwise, the system is nonhomogeneous.

Normal Form of a Homogenous System of Linear DEs with Constant Coefficients

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

Solutions to Systems of ODEs

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

Distinct Real Eigenvalues

Let be the real eigenvalues for the characteristic polynomial corresponding to the eigenvectors . Then the solution to the system of DEs, is

Example 01:

Solve the following system of DEs.


  1. Rewrite system as:

  2. Using technology, find the eigenvalues and corresponding eigenvectors of the coefficient matrix:


    1. 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.
    2. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.
    3. The eigenvector and corresponding eigenvalues will be returned in order.
  3. Using technology, the eigenvectors of the coefficient matrix are:

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

Investigation 01

Use matrix operations to solve the following systems of DEs:


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