Nonhomogeneous Linear Systems

Nonhomogeneous Linear SystemsExpected Educational ResultsNonhomogeneous Linear SystemsNormal Form of a System of Linear DEsNormal Form of a Nonhomogenous System of Linear DEs with Constant CoefficientsUndetermined CoefficientsParticular SolutionInvestigation 01Variation of ParametersMethodInvestigtion 02CC BY-NC-SA 4.0

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

Expected Educational Results

Nonhomogeneous Linear Systems

Normal Form of a System of Linear DEs

where ,

, and

is an matrix.

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

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

where are all constants, , and are polynomials, exponential functions, sines and cosines, sums and products of these functions.

Undetermined Coefficients

Particular Solution

Example 01:

Solve the following system of DEs.

Solution:

  1. Rewrite system as:

  2. Using technology, find the homogeneous solution to the system of DEs is\newline

  3. Using Method of Undetermined Coefficients, identify the linearly independent particular solution to the system of DEs:

  4. Find the values for and by substituting into the system of DEs:

    1. Rewrite as:
    2. Simplify to:\newline and and solve for unknown coefficients: and .
    3. Thus,
  5. The general solution is .

NOTE: See CPT_12_Method_Undetermined_Coefficients.pdf for a refresher on the Method of Undetermined Coefficients. Keep in mind the coefficients needed for are vectors.

Investigation 01

Solve the following systems of DEs:

Variation of Parameters

NOTE: This section is not part of the syllabus and you will not be tested on this method.

See 2652_INV_13_Variation_Of_Parameters.pdf for a refresher on the method of Variation of Parameters.

Method
  1. Find the fundamental solution set of linearly independent solutions, , where are column vectors.
  2. Let
  3. Evaluate

Investigtion 02

Use the Method of Variation of Parameters 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