The Method of Undetermined Coefficients
Consider the n-th order non-homogeneous equation with constant coefficients
a
0
y
n
( )
+
a
1
y
n
!
1
(
)
+
.....
+
a
n
!
1
"
y
+
a
n
y
=
b(x)
where a
0
, a
1
, …, a
n
are constant and b(x) is a non-constant function of x.
Let y
p
(x) be a particular solution of the non-homogeneous equation, containing no
arbitrary constants.
Let y
c
(x) = c
1
f
1
(x) + c
2
f
2
(x) + … + c
n
f
n
(x) where c
1
, c
2
, … , c
n
are arbitrary constants be
the general solution of the corresponding homogeneous equation
a
0
y
n
( )
+
a
1
y
n
!
1
(
)
+
.....
+
a
n
!
1
"
y
+
a
n
y
=
0
.
Then, the general solution of the equation can be expressed as:
y(x) = y
p
(x) + y
c
(x)
In this section, we will discuss how to find the particular solution y
p
(x).
Remark
: The kind of functions b(x) for which the method of undetermined coefficients
applies are actually quite restricted.
Definition
: A function f(x) is a
UC function
if it is either:
1) x
n
, where n is an integer n
≥
0.
2) e
ax
, where a is a constant different from 0.
3) sin(bx + c), where b and c are constant and b
≠
0.
4) cos(bx + c), where b and c are constant and b
≠
0.
5) any function that is a finite product of two or more functions of those four types.
Examples
:
x
3
, e
-4x
, sin(5x + 7), x
2
e
6x
, x cos(3x), e
5x
sin(3x – 7), cos(x) sin(2x), x
4
e
-5x
sin(2x).
Remark
: The method of undetermined coefficients applies when the non-homogeneous
term b(x), in the non-homogeneous equation is a linear combination of UC functions.
Remark
: Given a UC function f(x), each successive derivative of f(x) is either itself, a
constant multiple of a UC function or a linear combination of UC functions.
Definition
: Given a UC function f(x). We call
UC set of f(x)
, to the set of all UC
functions consisting of f(x) itself and all linearly independent functions of which the
successive derivatives of f(x) are either constant multiples or linear combinations.
Example
:
Find the UC set of f(x)
1) Given f(x) = x
5
,
its derivatives are:
f’(x) = 5x
4
, f’’(x) = 20x
3
, f’’’(x) 60x
2
, f
(4)
= 120x, f
(5)
= 120, f
(n)
(x) = 0, n>5.
The linearly independent functions of which the successive derivatives of f(x) are either
constant multiples or linear combinations are: x
4
, x
3
, x
2
, x, 1.
Then UC set of x
5
= {x
5
, x
4
, x
3
, x
2
, x, 1}.