Math 415
Homework 4
3.1.24
Determine the values of
α
for which all solutions of
y
00
+ (3

α
)
y
0

2(
α

1)
y
= 0 tend to zero as
t
→ ∞
; also determine the values of
α
for which all (nonzero) solutions become unbounded as
t
→ ∞
.
The characteristic equation for the given diﬀerential equation is 0 =
r
2
+(3

α
)
r

2(
α

1) = (
r
+2)(
r

(
α

1)).
So the general solution is
y
(
t
) =
C
1
e

2
t
+
C
2
e
(
α

1)
t
. The
e

2
t
term goes to zero as
t
approaches inﬁnity,
so we only need to concern ourselves with the second term. If
α

1
<
0 (i.e.
α <
1), then the solution will
approach zero as
t
→ ∞
. If
α >
1 then the solution will become unbounded as
t
→ ∞
(whether to positive
or negative inﬁnity depends on
C
2
).
3.2.25
Verify that
y
1
(
x
) =
x
and
y
2
(
x
) =
xe
x
are solutions of the diﬀerential equation
x
2
y
00

x
(
x
+ 2)
y
0
+
(
x
+ 2)
y
= 0 (with
x >
0). Do they constitute a fundamental set of solutions?
For
y
1
we get:
x
2
·
0

x
(
x
+ 2)
·
1 + (
x
+ 2)
x
= 0
.
And for
y
2
we get:
x
2
(
xe
x
+ 2
e
x
)

x
(
x
+ 2)(
xe
x
+
e
x
) + (
x
+ 2)
xe
x
= (
x
3
+ 2
x
2

x
3

x
2

2
x
2

2
x
+
x
2
+ 2
x
)
e
x
= 0
.
So both are solutions to the diﬀerential equation.
To see that they form a fundamental set of solutions, we check the Wronskian of the two:
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