Method of Undetermined Coefficients: Example
Find a particular solution to the
equation
Solution: Let us follow these steps:

 (1)
 First, we notice that the conditions are satisfied to invoke
the method of undetermined coefficients.
 (2)
 We split the equation into the following three equations:
 (3)
 The root of the characteristic equation are
r=1 and r=4.
 (4.1)
 Particular solution to Equation (1):

 Since , and , then ,
which is not one of the roots. Then s=0.

 The particular solution is given as

 If we plug it into the equation (1), we get
,
which implies A = 1/2, that is,
 (4.2)
 Particular solution to Equation (2):

 Since , and , then ,
which is not one of the roots. Then s=0.

 The particular solution is given as

 If we plug it into the equation (2), we get
,
which implies
Easy calculations give
,
and
,
that is
 (4.3)
 Particular solution to Equation (3):

 Since , and , then
which is one of the roots. Then s=1.

 The particular solution is given as

 If we plug it into the equation (3), we get
,
which implies
,
that is
 (5)
 A particular solution to the original equation is
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Author: Mohamed
Amine Khamsi
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