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 ODEsComplex EigenvaluesInvestigation 02Investigation 03CC 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 nonhomogenous.

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.

Complex Eigenvalues

Let be the complex conjugate eigenvalues for the characteristic polynomial corresponding to the complex eigenvector . Then the solution to the system of DEs, is

where is the real part of the solution and is the imaginary part of the solution.

NOTE: Only one eigenvalue and its corresponding eigenvector is needed.

Example 04:

Solve the following system of DEs.

Solution:

  1. Rewrite system as:

  2. Using technology, the eigenvalues of the coefficient matrix are

  3. Using technology, the eigenvectors of the coefficient matrix are:

    • For ,
    • For ,
  4. Find the fundamental solution set, expand using Euler's formula (See Question 1 in CPT_11_Complex_Solutions.pdf.):

  5. The solution to the system: which is equivalent to

Investigation 02

For the above example, use the other eigenvalue and corresponding eigenvector and verify the equivalence of the solutions.

Investigation 03

Use matrix operations to solve the following systems of DEs:



CC BY-NC-SA 4.0

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