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separation and comparison theorems

separation and comparison theorems - SEPARATION AND...

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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
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