SEPARATION AND COMPARISON THEOREMS
KIAM HEONG KWA
Recall that a
zero
of a function is a point where its value is zero. As
an application of the Wronskian, we use it to “separate” and “compare”
zeros of solutions to secondorder linear homogeneous equations.
Theorem 1
(Sturm Separation Theorem)
.
[Theorem 7, p. 47]
[1]
Let
φ
(
t
)
and
ψ
(
t
)
be linearly independent solutions of
(1)
y
00
+
p
(
t
)
y
0
+
q
(
t
)
y
= 0
,
where
p
(
t
)
and
q
(
t
)
are continuous functions, on an open interval
I
.
The zeros of
φ
(
t
)
and
ψ
(
t
)
occur alternately in the sense that each
must vanish somewhere between any two successive zeros of another.
Proof.
In view of the linear independence of
φ
(
t
) and
ψ
(
t
),
W
(
φ, ψ
)(
t
)
6
=
0 on
I
. In particular, it always has the same sign so that
(2)
W
(
φ, ψ
)(
t
1
)
W
(
φ, ψ
)(
t
2
)
>
0
for any two points
t
1
, t
2
∈
I
.
Let
τ
1
and
τ
2
be two successive zeros of
ψ
(
t
). By the uniqueness of
solutions with respect to the initial conditions,
ψ
0
(
τ
1
)
6
= 0 and
ψ
0
(
τ
2
)
6
=
0; otherwise,
ψ
(
t
) vanishes identically.
If
ψ
0
(
τ
1
)
>
0, then
ψ
(
t
)
>
0
for
τ
1
< t < τ
2
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 Winter '11
 Kwa
 Differential Equations, Topology, Equations, Intermediate Value Theorem, Partial Differential Equations, Continuous function, Sturm Separation Theorem

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