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**:

**Section 4.4**– Nonhomogeneous Equations: The Method of Undetermined Coefficients**Section 4.5**– The Superposition Principle and Undetermined Coefficients Revisited

**Objective 12–1**: I can find particular solutions to nonhomogeneous equations.**Objective 12–2**: I can find general solutions to nonhomogeneous equations.**Objective 12–2**: I can solve higher-order nonhomogeneous IVPs.

, , .

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

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

Definition: General Solution

. is an acceptable notation for the general solution.

**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.

Find the homogeneous solution, .

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

:

- is not a root of the characteristic equation:
- is a root of the characteristic equation:
- is a double root of the characteristic equation:

, :

- is not a root of the characteristic equation:
- is a root of the characteristic equation:

Find coefficients by finding , , etc. and substituting into the DE and solving for the coefficients.

Write the general solution,

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 , , .

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

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

- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,

Solve the following DEs:

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