Method of Undetermined Coefficient or Guessing Method
Method of Undetermined Coefficient or Guessing Method
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
polynomial functions.
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
,
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
where
, 
or 
, where 
is a polynomial function
with degree n, then split this equation into N equations
; 
, and
find its roots;
. Compare this
number to the roots of the characteristic equation found in previous step.
In other words, s measures how many times is not one of the roots, then set
s = 0;
is one of the two distinct
roots, set s = 1;
is equal to both root (which
means that the characteristic equation has a double root), set s=2.
is
a root of the characteristic equation;
and 
by plugging 
into the
equation
are found, then the
particular solution of the original equation is
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Author: Mohamed Amine Khamsi