Chapter 4
For
t
= 0,
y
1
(0) = 3
·
0
2
= 0 and
y
2
(0) = 0. The latter follows from the fact that onesided
derivatives of
y
2
(
t
), 3
t
2
and
−
3
t
2
, are both zero at
t
= 0. Also,
y
1
(0) =
y
2
(0) = 0. Hence
W
[
y
1
, y
2
](0) =
0
0
0
0
= 0
,
and so
W
[
y
1
, y
2
](
t
)
≡
0 on (
−∞
,
∞
). 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
+
by
+
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
. Thus, if
C
= 0, then
W
[
y
1
, y
2
](
t
) = 0 for any
t
in (
−∞
,
∞
), because the exponential function,
e
−
bt/a
, is never zero.
For
C
= 0,
W
[
y
1
, y
2
](
t
)
≡
0 on (
−∞
,
∞
).
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
3
t
2
,
is a polynomial of degree at most two and so can have at most two real roots, unless it is
a zero polynomial, i.e., has all zero coeﬃcients. Therefore, the above linear combination
vanishes on (
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 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations, Derivative, Elementary algebra

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