separation and comparison theorems - SEPARATION AND...

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 second-order 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