Explanation Given: The function ϕ is the solution to the initial value problem d y / d x = f ( x , y ) , y ( x 0 ) = y 0 ψ is the solution to d y / d x = F ( x , y ) , y ( x 0 ) = y ˜ 0 Approach: 1) Suppose that u ( x ) > 0 , v ( x ) ≥ 0 are continuous functions on [ a , b ] . If v ( x ) ≤ c + ∫ a x u ( t ) v ( t ) d t for some constant c ≥ 0 . Then, v ( x ) ≤ c e ∫ a x u ( t ) d t Here, a ≤ x ≤ b . 2) The solution to the initial value problem y ′ = f ( x , y ) , y ( x 0 ) = y 0 is given by y ( x ) = y 0 + ∫ x 0 x f ( s , y ( s , y 0 ) ) d s Calculation: The solutions ϕ and ψ are given by ϕ ( x ) = y 0 + ∫ x 0 x f ( s , ϕ ( s , y 0 ) ) d s ... ( 1 ) ψ ( x ) = y 0 + ∫ x 0 x F ( s , ψ ( s , y ˜ 0 ) ) d s ... ( 2 ) for all | x − x 0 | ≤ h . Subtract equation ( 2 ) from equation ( 1 ) and take modulus | ϕ ( x ) − ψ ( x ) | = | y 0 + ∫ x 0 x f ( s , ϕ ( s , y 0 ) ) d s − y ˜ 0 − ∫ x 0 x F ( s , ϕ ( s , y ˜ 0 ) ) d s | = | y 0 − y ˜ 0 + ∫ x 0 x f ( s , ϕ ( s , y 0 ) ) d s − ∫ x 0 x f ( s , ψ ( s , y 0 ) ) d s + ∫ x 0 x f ( s , ψ ( s , y 0 ) ) d s − ∫ x 0 x F ( s , ψ ( s , y ˜ 0 ) ) d s | ≤ | y 0 − y ˜ 0 | + | ∫ x 0 x [ f ( s , ϕ ( s , y 0 ) ) − f ( s , ψ ( s , y ˜ 0 ) ) ] d s | + | ∫ x 0 x [ f ( s , ψ ( s , y 0 ) ) − F ( s , ψ ( s , y ˜ 0 ) ) ] d

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Question

Chapter 13.4, Problem 16E

To determine

To show:

For |x−x0|≤h,

|ϕ(x)−ψ(x)|≤{|y0−y˜0|+εh}eLh.

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Chapter 13.4, Problem 14E

Chapter 13.RP, Problem 1RP

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