Method of Undetermined Coefficients

A method to find the particular solution of a higher-order linear differential equation. To preform this method one must first have differential equation with the dependent variable ( in the example below) all on one side.

First the general solution of the homogenous form of the equation is found by setting the left side equal to zero (). Solving this equation as a homogeneous differential equation yields the general solution, referred to as moving forward (h for homogenous).

The right side of the equation is then considered on its own () and is matched to the case (or cases) that describes it best in the table below resulting in (from cases 2 and 5). If has terms that are duplicates of those in then that term must be multiplied by where is the smallest possible integer that eliminates the duplication (See situation 2 below). refers to the particular solution, is from and is from .

The derivative of is taken to as high an derivative order as in the original equation. The derivatives (and original particular solution) are plugged into the original differential equation so that the coefficients (A,B,C, and D) can be found.

The general and particular solutions are then summed to get a solution ().

See W3L1 - Method of Undetermined Coefficients for more info and examples.

Situations

Situation 1: No function in the assumed particular solution is a solution of the associated homogeneous differential equation. See Picking the Right Situation for more information on what this means.

#		Form of
1.	(any constant)
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.

Situation 2: If any contains terms that duplicate terms in then that term must be multiplied where is the smallest positive integer that eliminates that duplication. See W3L1 - Method of Undetermined Coefficients for examples.

Picking the Right Situation

Given an example differential equation and homogeneous solution ():

Example 1: If then the particular solution is (case 6). None of the functions in the homogenous solution are in the particular solution so we continue with situation 1 above.

Example 2: If then the particular solution is (case 7). Because the function appears in the homogenous solution the particular solution must be multiplied by . t might need to be multiplied by a higher to properly cancel?