Method of Undetermined Coefficient or Guessing Method

This method is based on a guessing technique. That is, we will guess the form of tex2html_wrap_inline55 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

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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

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where a, b and c are constants and

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where tex2html_wrap_inline73 is a polynomial function with degree n. Then a particular solution tex2html_wrap_inline55 is given by

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where

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where the constants tex2html_wrap_inline79 and tex2html_wrap_inline81 have to be determined. The power s is equal to 0 if tex2html_wrap_inline87 is not a root of the characteristic equation. If tex2html_wrap_inline87 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

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where tex2html_wrap_inline97 are of the forms cited above, then we split the original equation into N equations

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then find a particular solution tex2html_wrap_inline101 . A particular solution to the original equation is given by

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

Example

[Differential Equations] [First Order D.E.] [Second Order D.E.]
[Geometry] [Algebra] [Trigonometry ]
[Calculus] [Complex Variables] [Matrix Algebra]

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Author: Mohamed Amine Khamsi

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