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**:

**Section 9.4**– 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**Section 9.5**-- Nonhomogeneous Linear Systems with Constant Coefficients

**Objective 18–1**: I can non-homogeneous linear systems using eigenvalues and eigenvectors.

where ,

, and

is an matrix.

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

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

**Example 01**:

Solve the following system of DEs.

**Solution**:

Rewrite system as:

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

Using

*Method of Undetermined Coefficients*, identify the**linearly independent**particular solution to the system of DEs:Find the values for and by substituting into the system of DEs:

- Rewrite as:
- Simplify to:\newline and and solve for unknown coefficients: and .
- Thus,

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.

Solve the following systems of DEs:

**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*.

- Find the fundamental solution set of linearly independent solutions, , where are column vectors.
- Let
- Evaluate

Use the Method of Variation of Parameters to solve the following systems of DEs:

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