Variation of ParametersExpected Educational ResultsVariation of ParametersDefinition: Particular SolutionDefinition: Variation of ParametersDefinition: General SolutionMethod of Variation of ParametersDerivation of the Method of Variation of ParametersInvestigation 01CC BY-NC-SA 4.0

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

**Section 4.6**-- Variation of Parameters

**Objective 13–1**: I understand when variation of parameters is required to find particular solutions to nonhomogeneous equations.**Objective 13–1**: I can use variation of parameters to find particular solutions to nonhomogeneous equations.

A solution to a nonhomogeneous DE is called the **particular solution**, .

**Variation of parameters** is a more general method to find .

Variation of parameters is used for nonhomogeneous solutions when contains factors or terms other than , , , and .

.

**NOTE**: The terms determined by the method of variation of parameters **must** be linearly independent of the terms in .

Find two linearly independent solutions to the corresponding homogeneous equation.

Let , for some functions and .

Solve for :

Solve for :

The particular solution is .

Write the general solution, .

Conditions

- Suppose is the homogeneous solution to , .
- Let for some functions and .
- Let [this condition is a simplifying assumption].

We need to show Equation (\ref{eq:v1}) and Equation (\ref{eq:v2}) above are consequences of the two conditions.

Find . Use the simplifying assumption to simplify . Explain why the simplifying assumption is helpful.

Find .

Substitute , , and into .

Expand the expression on the left side of the equation.

Since we are interested in finding functions, and , factor the expression using common factors, and .

What do you notice about the factors multiplying and ? Explain.

Divide through by the coefficient .

We now have two equations with two unknowns, and :

- The equation from Step vii.
- The simplifying assumption.

Solve for and .

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**Last Modified**: Thursday, 8 October 2020 7:38 EDT