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Math / Differential Equations and Their Applications: An Introduction to Applied Mathematics 3 / Chapter 1.10 / Problem 16

Differential Equations and Their Applications: An Introduction to Applied Mathematics | 3rd Edition | ISBN: 9780387908069 | Authors: M. Braun

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Consider the initial-value problem and let R be the rectangle 0 < t < a, - b < y < b

Chapter 1, Problem 16

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QUESTION:

Consider the initial-value problem and let R be the rectangle 0 < t < a, - b < y < b. (a) Show that the solution y(t) of (*) (b) Show that the maximum value of b/(a2+ b2), for a fixed, is 1/2a.(c) Show that a = min(a, $a) is largest when a = 1 / fi . (d) Conclude that the solution y (t) of (*) exists for 0 < t < 1 / fi

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QUESTION:

Consider the initial-value problem and let R be the rectangle 0 < t < a, - b < y < b. (a) Show that the solution y(t) of (*) (b) Show that the maximum value of b/(a2+ b2), for a fixed, is 1/2a.(c) Show that a = min(a, $a) is largest when a = 1 / fi . (d) Conclude that the solution y (t) of (*) exists for 0 < t < 1 / fi

ANSWER:

Step 1 of 7

(a).

Assume the given conditions. We will heavily use Theorem 2 and Theorem 2’ to show what is asked of us. Let’s start by maximum:

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