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Chapter 4
For
t
=0
,
y
0
1
(0) = 3
·
0
2
=0and
y
0
2
(0) = 0. The latter follows from the fact that one-sided
derivatives of
y
2
(
t
), 3
t
2
and
−
3
t
2
, are both zero at
t
=0
. A
lso
,
y
1
(0) =
y
2
(0) = 0. Hence
W
[
y
1
,y
2
](0) =
±
±
±
±
±
00
00
±
±
±
±
±
=0
,
and so
W
[
y
1
,y
2
](
t
)
≡
0on(
−∞
,
∞
). This result does not contradict part (b) in Prob-
lem 34 because these functions are not a pair of solutions to a homogeneous linear
equation with constant coeﬃcients.
37.
If
y
1
(
t
)and
y
2
(
t
) are solutions to the equation
ay
0
+
by
0
+
c
= 0, then, by Abel’s formula,
W
[
y
1
,y
2
](
t
)=
Ce
−
bt/a
,where
C
is a constant depending on
y
1
and
y
2
.Thu
s
,i
f
C
6
=0
,then
W
[
y
1
,y
2
](
t
)
6
=0forany
t
in (
−∞
,
∞
), because the exponential function,
e
−
bt/a
, is never zero.
For
C
=0
,
W
[
y
1
,y
2
](
t
)
≡
0on(
−∞
,
∞
).
39. (a)
A linear combination of
y
1
(
t
)=1,
y
2
(
t
)=
t
,and
y
3
(
t
)=
t
2
,
C
1
·
1+
C
2
·
t
+
C
3
·
t
2
=
C
1
+
C
2
t
+
C

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