The Method of Undetermined Coefficients

The Method of Undetermined CoefficientsExpected Educational ResultsMethod of Undetermined CoefficientsNonhomogeneous Linear EquationsDefinition: Nonhomogeneous Linear EquationsTheorem: Superposition PrincipleDefinition: Particular SolutionMethod of Undetermined Coefficients – ProcedureExistence and UniquenessInvestigation 01Investigation 02Investigation 03CC BY-NC-SA 4.0

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

Expected Educational Results

Method of Undetermined Coefficients

Nonhomogeneous Linear Equations

Definition: Nonhomogeneous Linear Equations

, , .

Theorem: Superposition Principle

Let be a solution to and be a solution to Then for any constants and , the function is a solution to the DE

Definition: Particular Solution

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

Definition: General Solution

. is an acceptable notation for the general solution.

Method of Undetermined Coefficients – Procedure

NOTE: The terms in must be linearly independent of the terms in . This method is valid only when consists of polynomials, exponentials, sines, cosines, or sums and products of these functions.

  1. Find the homogeneous solution, .

  2. Let be a polynomial of degree , be a polynomial of degree , .

  3. :

    1. is not a root of the characteristic equation:
    2. is a root of the characteristic equation:
    3. is a double root of the characteristic equation:
  4. , :

    1. is not a root of the characteristic equation:
    2. is a root of the characteristic equation:
  5. Find coefficients by finding , , etc. and substituting into the DE and solving for the coefficients.

  6. Write the general solution,

Existence and Uniqueness

THeorem: Existence and Uniqueness of Nonhomogeneous DEs

For any real numbers , , , , , and , suppose is a particular solution to on some interval containing and suppose and are linearly independent solutions to the homogeneous equation in the interval, then there exists a unique solution in the interval to the IVP , , .

Investigation 01

Suppose is the homogeneous solution to . Let be the particular solution to the DE. Show is the general solution to the DE.

Investigation 02

Suppose the following are roots, , to the characteristic equation and is the nonhomogeneous function to the ode. Find the general solution, .

  1. ,
  2. ,
  3. ,
  4. ,
  5. ,
  6. ,
  7. ,
  8. ,
  9. ,

Investigation 03

Solve the following DEs:

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Last Modified: Monday, 5 October 2020 12:22 EDT