SECTION 17.5:
Linear Differential Equations with Constant CoefFcients
919
Exercises 17.4
1.
Show that
y
=
e
x
is a solution of
y
00
−
3
y
0
+
2
y
=
0, and
Fnd the general solution of this DE.
2.
Show that
y
=
e
−
2
x
is a solution of
y
00
−
y
0
−
6
y
=
0, and
Fnd the general solution of this DE.
3.
Show that
y
=
x
is a solution of
x
2
y
00
+
2
xy
0
−
2
y
=
0on
the interval
(
0
,
∞
)
, and Fnd the general solution on this
interval.
4.
Show that
y
=
x
2
is a solution of
x
2
y
00
−
3
0
+
4
y
=
the interval
(
0
,
∞
)
, and Fnd the general solution on this
interval.
5.
Show that
y
=
x
is a solution of the differential equation
x
2
y
00
−
(
2
x
+
x
2
)
y
0
+
(
2
+
x
)
y
=
0, and Fnd the general
solution of this equation.
6.
Show that
y
=
x
−
1
/
2
cos
x
is a solution of the Bessel
equation with
ν
=
1
/
2:
x
2
y
00
+
0
+
±
x
2
−
1
4
²
y
=
0
.
±ind the general solution of this equation.
First-order systems
7.
Asystemof
n
Frst-order, linear, differential equations in
n
unknown functions
y
1
,
y
2
,
···
,
y
n
is written
y
0
1
=
a
11
(
x
)
y
1
+
a
12
(
x
)
y
2
+···+
a
1
n
(
x
)
y
n
+
f
1
(
x
)
y
0
2
=
a
21
(
x
)
y
1
+
a
22
(
x
)
y
2
a
2
n
(
x
)
y
n
+
f
2
(
x
)
.
.
.
y
0
n
=
a
n
1
(
x
)
y
1
+
a
n
2
(
x
)
y
2
a
nn
(
x
)
y
n
+
f
n
(
x
).
Such a system is called an
n
×
n
frst-order linear system
and can be rewritten in vector-matrix form as
y
0
=
A
(
x
)
y
+
±
(
x
)
,where
y
(
x
)
=
⎛
⎝
y
1
(
x
)
.
.
.
y
n
(
x
)
⎞
⎠
,
±
(
x
)
=
⎛
⎝
f
1
(
x
)
.
.
.
f
n
(
x
)
⎞
⎠
,
A
(
x
)
=
⎛
⎝
a
11
(
x
)
a
1
n
(
x
)
.
.
.
.
.
.
.
.
.
a
n
1
(
x
)
a
(
x
)
⎞
⎠
.
Show that the second-order, linear equation
y
00
+
a
1
(
x
)
y
0
+
a
0
(
x
)
y
=
f
(
x
)
can be transformed into a
2
×
2 Frst-order system with
y
1
=
y
and
y
2
=
y
0
having
A
(
x
)
=
±
01
−
a
0
(
x
)
−
a
1
(
x
)
²
,
±
(
x
)
=
±
0
f
(
x
)
²
.
8.
Generalize Exercise 7 to transform an
n
th-order linear
equation
y
(
n
)
+
a
n
−
1
(
x
)
y
(
n
−
1
)
+
a
n
−
2
(
x
)
y
(
n
−
2
)
a
0
(
x
)
y
=
f
(
x
)
into an
n
×
n
Frst-order system.
9.
If
A
is an
n
×
n
constant matrix, and if there exists a scalar
λ
and a nonzero constant vector
v
for which
A
v
=
λ
v
,show
that
y
=
C
1
e
λ
x
v
is a solution of the homogeneous system
y
0
=
A
y
.
10.
Show that the determinant
³
³
³
³
2
−
λ
1
23
−
λ
³
³
³
³
is zero for two
distinct values of
λ
. ±or each of these values Fnd a nonzero
vector
v
that satisFes the condition
±
21
²
v
=
λ
v
.Hence
solve the system
y
0
1
=
2
y
1
+
y
2
,
y
0
2
=
2
y
1
+
3
y
2
.
17.5
Linear Differential Equations with Constant CoefFcients
A differential equation of the form
ay
00
+
by
0
+
cy
=
0
,(
∗
)
where
a
,
b
,and
c
are constants and
a
6=
0, is said to be a
linear, homogeneous,
second-order equation with constant coe±fcients
.
A thorough discussion o± techniques ±or solving such equations, together
with examples, exercises, and applications to the study o± simple and
damped harmonic motion, can be ±ound in Section 3.7; we will not repeat
that discussion here. I± you have not studied it, please do so now.