Method of Undetermined Coefficient or Guessing Method

This method is based on a guessing technique. That is, we will guess the form of and then plug it in the equation to find it. However, it works only under the following two conditions:

Condition 1: the associated homogeneous equations has constant coefficients;

Condition 2: the nonhomogeneous term g(x) is a special form

where P(x) and L(x) are polynomial functions.
Note that we may assume that g(x) is a sum of such functions (see the remark below for more on this).

Assume that the two conditions are satisfied. Consider the equation

where a, b and c are constants and

where is a polynomial function with degree n. Then a particular solution is given by

where

,

where the constants and have to be determined. The power s is equal to 0 if is not a root of the characteristic equation. If is a simple root, then s=1 and s=2 if it is a double root.

Remark: If the nonhomogeneous term g(x) satisfies the following

where are of the forms cited above, then we split the original equation into N equations

then find a particular solution . A particular solution to the original equation is given by

Summary:Let us summarize the steps to follow in applying this method:

(1)

First, check that the two conditions are satisfied;

(2)

If the equation is given as

,

where or , where is a polynomial function with degree n, then split this equation into N equations

;

(3)

Write down the characteristic equation , and find its roots;

(4)

Write down the number . Compare this number to the roots of the characteristic equation found in previous step.

(4.1)

If is not one of the roots, then set s = 0;

(4.2)

If is one of the two distinct roots, set s = 1;

(4.3)

If is equal to both root (which means that the characteristic equation has a double root), set s=2.

In other words, s measures how many times is a root of the characteristic equation;

(5)

Write down the form of the particular solution



where



(6)

Find the constants and by plugging into the equation



(7)

Once all the particular solutions are found, then the particular solution of the original equation is

Example

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