21
Method of Undetermined Coefficients
(aka: Method of Educated Guess)
In this chapter, we will discuss one particularly simple-minded, yet often effective, method
for finding particular solutions to nonhomogeneous differential equations. As the above title
suggests, the method is based on making “good guesses” regarding these particular solutions.
And, as always, “good guessing” is usually aided by a thorough understanding of the problem
(the ‘education’), and usually works best if the problem is simple enough. Fortunately, you have
had the necessary education, and a great many nonhomogeneous differential equations of interest
are sufficiently simple.
As usual, we will start with second-order equations, and then observe that everything devel-
oped also applies, with little modification, to similar nonhomogeneous differential equations of
any order.
21.1
Basic Ideas
Suppose we wish to find a particular solution to a nonhomogeneous second-order differential
equation
ay
′′
+
by
′
+
cy
=
g
.
If
g
is a relatively simple function and the coefficients —
a
,
b
and
c
— are constants, then,
after recalling what the derivatives of various basic functions look like, we might be able to make
a good guess as to what sort of function
y
p
(
x
)
yields
g
(
x
)
after being plugged into the left side
of the above equation. Typically, we won’t be able to guess exactly what
y
p
(
x
)
should be, but
we can often guess a formula for
y
p
(
x
)
involving specific functions and some constants that can
then be determined by plugging the guessed formula for
y
p
(
x
)
into the differential equation and
solving the resulting algebraic equation(s) for those constants (provided the initial ‘guess’ was
good).
!
◮
Example 21.1:
Consider
y
′′
−
2
y
′
−
3
y
=
36
e
5
x
.
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418
Method of Educated Guess
Since all derivatives of
e
5
x
equal some constant multiple of
e
5
x
, it should be clear that, if we
let
y
(
x
)
=
some multiple of
e
5
x
,
then
y
′′
−
2
y
′
−
3
y
=
some other multiple of
e
5
x
.
So let us let
A
be some constant “to be determined”, and try
y
p
(
x
)
=
Ae
5
x
as a particular solution to our differential equation:
y
p
′′
−
2
y
p
′
−
3
y
p
=
36
e
5
x
equal1⇒
bracketleftbig
Ae
5
x
bracketrightbig
′′
−
2
bracketleftbig
Ae
5
x
bracketrightbig
′
−
3
bracketleftbig
Ae
5
x
bracketrightbig
=
36
e
5
x
equal1⇒
bracketleftbig
25
Ae
5
x
bracketrightbig
−
2
bracketleftbig
5
Ae
5
x
bracketrightbig
−
3
bracketleftbig
Ae
5
x
bracketrightbig
=
36
e
5
x
equal1⇒
25
Ae
5
x
−
10
Ae
5
x
−
3
Ae
5
x
=
36
e
5
x
equal1⇒
12
Ae
5
x
=
36
e
5
x
equal1⇒
A
=
3
.
So our “guess”,
y
p
(
x
)
=
Ae
5
x
, satisfies the differential equation only if
A
=
3
. Thus,
y
p
(
x
)
=
3
e
5
x
is a particular solution to our nonhomogeneous differential equation.
In the next section, we will determine the appropriate “first guesses” for particular solutions
corresponding to different choices of
g
in our differential equation. These guesses will involve
specific functions and initially unknown constants that can be determined as we determined
A
in the last example. Unfortunately, as we will see, the first guesses will sometimes fail. So we

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