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:
where P(x) and L(x) are
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 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:
where or , where is a polynomial function
with degree n, then split this equation into N equations
In other words, s measures how many times is
a root of the characteristic equation;
where or , where is a polynomial function with degree n, then split this equation into N equations
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Author: Mohamed Amine Khamsi