Explanation Given: The function ϕ ( x ) is a nontrivial solution to y ″ + x − 2 y ′ + ( 4 − e − x ) y = 0 on ( 0 , ∞ ) . Approach: Suppose that ϕ is a nontrivial solution to the differential equation d d x ( p d y d x ) + q y + λ r y = 0 on ( a , b ) . Here, p , p ′ , q and r are continuous [ a , b ] ; and p , q are positive in [ a , b ] . The distance between two consecutive zeroes x 1 and x 2 of ϕ is given by π p m q M + λ r M ≤ x 2 − x 1 ≤ π p M q m + λ r m Here, the functions f m and f M denote the minimum and maximum values of f respectively on [ a , b ] . Calculation: From given equation, p = 1 q = 4 λ = − 1 r = e − x These values satisfy all the conditions of the hypothesis of above result

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 11.8, Problem 9E

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The distance between two consecutive zeroes of ϕ(x) when x is large.

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Chapter 11.8, Problem 8E

Chapter 11.8, Problem 10E

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Chapter 11 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - Prob. 3E Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - Prob. 7E Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - Prob. 9E Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...

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