Variation of Parameters

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:

Expected Educational Results

Variation of Parameters

Definition: Particular Solution

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

Definition: Variation of Parameters

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 .

Definition: General Solution


Method of Variation of Parameters

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

  1. Find two linearly independent solutions to the corresponding homogeneous equation.

  2. Let , for some functions and .

  3. Solve for :

  4. Solve for :

  5. The particular solution is .

  6. Write the general solution, .

Derivation of the Method of Variation of Parameters

Investigation 01

  1. Conditions

    1. Suppose is the homogeneous solution to , .
    2. Let for some functions and .
    3. Let [this condition is a simplifying assumption].
  2. We need to show Equation (\ref{eq:v1}) and Equation (\ref{eq:v2}) above are consequences of the two conditions.

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

    2. Find .

    3. Substitute , , and into .

    4. Expand the expression on the left side of the equation.

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

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

    7. Divide through by the coefficient .

    8. We now have two equations with two unknowns, and :

      1. The equation from Step vii.
      2. The simplifying assumption.
    9. Solve for and .


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