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 formwhere

*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).

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.

*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

**
**

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