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Method of Undetermined Coefficients or Guessing Method
As for the second order case, we have to satisfy two conditions, one of which is already satisfied since we assumed that our equation has constant coefficients. The second condition has to do with the nonhomogeneous term g(x). Indeed, in order to use the undetermined coefficients method, g(x) should be one of the elementary form
where
Find its roots and (specially) their multiplicity. Note that it will
help strongly if you factorize this equation. This way you get the
roots and their multiplicity.
where T(x) and R(x) are two polynomial functions with degree(T) =
degree(R) = degree(P). So if the degree of P is m, there are
2m+2 coefficients to be determined.
is a polynomial function. For a more general case, see
the remark below. In order to guess the form of
the particular solution we follow the steps:
(which you
generate from g(x)). Then
is not one of the root of the
characteristic equation, then set s=0;
is one of the root of the
characteristic equation, then s is its multiplicity.
into the equation (NH) to determine the
coefficients of T and R.
.
Remark. The undetermined coefficients method can still be used if
where 
has the elementary form described above. Indeed, we
need (as we did for the second order case) to split the equation
(NH) into m equations. Find the particular solution to each one,
then add them to generate the particular solution of the original equation.
Example: Find a particular solution of
Solution. Let us follow the steps:
We have the factorization
where A and B are to be determined. We will omit the detail of
the calculations. We get A = -1/8 and B=0. Therefore, we have
where A and B are to be determined. We will omit the detail of
the calculations. We get A = 0 and B=-3/5. Therefore, we have
. Therefore the
roots are 0,2,-2 and they are all simple.
which a simple root. Then s
= 1.
which is not a root. Then s
= 0.
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