# 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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