{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

separation and comparison theorems

# separation and comparison theorems - SEPARATION AND...

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}