SECTION 17.6:
Nonhomogeneous Linear Equations
923
The corresponding powers
x
r
can be expressed in real form in a manner similar to that
used for constant coefficient equations; we have
x
α
±
i
β
=
e
(α
±
i
β)
ln
x
=
e
α
ln
x
cos
(β
ln
x
)
±
i
sin
(β
ln
x
)
=
x
α
cos
(β
ln
x
)
±
ix
α
sin
(β
ln
x
).
Accordingly, the Euler equation has the general solution
y
=
C
1

x

α
cos
(β
ln

x

)
+
C
2

x

α
sin
(β
ln

x

).
Example 4
Solve the DE
x
2
y
−
3
xy
+
13
y
=
0.
Solution
The DE has the auxiliary equation
r
(
r
−
1
)
−
3
r
+
13
=
0, that is,
r
2
−
4
r
+
13
=
0, which has roots
r
=
2
±
3
i
. The DE, therefore, has the general
solution
y
=
C
1
x
2
cos
(
3 ln

x

)
+
C
2
x
2
sin
(
3 ln

x

).
Exercises 17.5
Exercises involving the solution of secondorder, linear,
homogeneous equations with constant coefficients can be found
at the end of Section 3.7.
Find general solutions of the DEs in Exercises 1–4.
1.
y
−
4
y
+
3
y
=
0
2.
y
(
4
)
−
2
y
+
y
=
0
3.
y
(
4
)
+
2
y
+
y
=
0
4.
y
(
4
)
+
4
y
(
3
)
+
6
y
+
4
y
+
y
=
0
5.
Show that
y
=
e
2
t
is a solution of
y
−
2
y
−
4
y
=
0
(where
denotes
d
/
dt
), and find the general solution of this
DE.
6.
Write the general solution of the linear, constantcoefficient
DE having auxiliary equation
(
r
2
−
r
−
2
)
2
(
r
2
−
4
)
2
=
0.
Find general solutions to the Euler equations in Exercises 7–12.
7.
x
2
y
−
xy
+
y
=
0
8.
x
2
y
−
xy
−
3
y
=
0
9.
x
2
y
+
xy
−
y
=
0
10.
x
2
y
−
xy
+
5
y
=
0
11.
x
2
y
+
xy
=
0
12.
x
2
y
+
xy
+
y
=
0
13.
3
Solve the DE
x
3
y
+
xy
−
y
=
0 in the interval
x
>
0.
17.6
Nonhomogeneous Linear Equations
We now consider the problem of solving the nonhomogeneoussecondorderdifferential
equation
a
2
(
x
)
d
2
y
dx
2
+
a
1
(
x
)
dy
dx
+
a
0
(
x
)
y
=
f
(
x
).
(
∗
)
We assume that two independent solutions,
y
1
(
x
)
and
y
2
(
x
)
, of the corresponding
homogeneous equation
a
2
(
x
)
d
2
y
dx
2
+
a
1
(
x
)
dy
dx
+
a
0
(
x
)
y
=
0
are known. The function
y
h
(
x
)
=
C
1
y
1
(
x
)
+
C
2
y
2
(
x
)
, which is the general solution
of the homogeneous equation, is called the
complementary function
for the nonho
mogeneous equation. Theorem 2 of Section 17.1 suggests that the general solution of
the nonhomogeneous equation is of the form
y
=
y
p
(
x
)
+
y
h
(
x
)
=
y
p
(
x
)
+
C
1
y
1
(
x
)
+
C
2
y
2
(
x
),
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924
CHAPTER 17
Ordinary Differential Equations
where
y
p
(
x
)
is any
particular solution
of the nonhomogeneous equation.
All we
need to do is find
one solution
of the nonhomogeneous equation, and we can write the
general solution.
There are two common methods for finding a particular solution
y
p
of the nonho
mogeneous equation
(
∗
)
:
1. the method of undetermined coefficients and
2. the method of variation of parameters.
The first of these hardly warrants being called a
method
; it just involves making
an educated guess about the form of the solution as a sum of terms with unknown
coefficients and substituting this guess into the equation to determine the coefficients.
This method works well for simple DEs, especially ones with constant coefficients.
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 Spring '08
 Graham
 general solution, Nonhomogeneous Linear Equations

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