Program Explanation Given information: The half-life for U 238 is 4.51 × 10 9 and the half-life for U 235 is 7.10 × 10 8 . The amount of two uranium isotopes U 238 and U 235 is same. At present the number of atoms for U 238 is 137.7 for each atom of U 235 . Calculation: Consider A is the initial number of atoms of both the uranium isotopes. Now consider B ( t ) is the number of number of atoms for U 238 and C ( t ) is the number of number of atoms for U 235 . The differential equation for U 235 is as follows, d B d t = − k 1 B Now use the method of separation of the variable to solve the above differential equation. d B d t = − k 1 B d B B = − k 1 d t Now integrate both sides. ∫ d B B = ∫ − k 1 d t ln B = − k 1 t + c 1 B = A e − k 1 t Therefore, the number of number of atoms for U 235 is as follows, B ( t ) = A e − k 1 t ........(1) Here, k 1 is the rate of disintegration of U 235 . The differential equation for U 238 is as follows, d C d t = − k 2 C Now use the method of separation of the variable to solve the above differential equation. d C d t = − k 2 C d C C = − k 2 d t Now integrate both sides. ∫ d B C = ∫ − k 2 d t ln C = − k 2 t + c 1 C = A e − k 2 t Therefore, the number of number of atoms for U 238 is as follows, C ( t ) = A e − k 2 t ........(2) Here, k 2 is the rate of disintegration of U 238

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

5th Edition

ISBN: 9780321816252

Author: C. Henry Edwards, David E. Penney, David Calvis

Publisher: PEARSON

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Recurrence relation

A recurrence relation or recursive relation is an equation that represents a function in terms of the values of its smaller inputs. Every recurrence relation T(n) is a recursive function of integer n and consists of a base case and a recursive case.

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Chapter 1.4, Problem 48P

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Program Description: Purpose of the problem is to find the age of the universe.

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Chapter 1.4, Problem 47P

Chapter 1.4, Problem 49P

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Chapter 1 Solutions

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

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Prob. 10P Ch. 1.3 - Prob. 11P Ch. 1.3 - Prob. 12P Ch. 1.3 - Prob. 13P Ch. 1.3 - Prob. 14P Ch. 1.3 - Prob. 15P Ch. 1.3 - Prob. 16P Ch. 1.3 - Prob. 17P Ch. 1.3 - Prob. 18P Ch. 1.3 - Prob. 19P Ch. 1.3 - Prob. 20P Ch. 1.3 - Prob. 21P Ch. 1.3 - Prob. 22P Ch. 1.3 - Prob. 23P Ch. 1.3 - Prob. 24P Ch. 1.3 - Prob. 25P Ch. 1.3 - Prob. 26P Ch. 1.3 - Prob. 27P Ch. 1.3 - Prob. 28P Ch. 1.3 - Verify that if c is a constant, then the function...Ch. 1.3 - Prob. 30P Ch. 1.3 - Prob. 31P Ch. 1.3 - Prob. 32P Ch. 1.3 - Prob. 33P Ch. 1.3 - (a) Use the direction field of Problem 5 to...Ch. 1.3 - Prob. 35P Ch. 1.4 - Prob. 1P Ch. 1.4 - Prob. 2P Ch. 1.4 - Prob. 3P Ch. 1.4 - Prob. 4P Ch. 1.4 - Prob. 5P Ch. 1.4 - Prob. 6P Ch. 1.4 - Prob. 7P Ch. 1.4 - Prob. 8P Ch. 1.4 - Prob. 9P Ch. 1.4 - Prob. 10P Ch. 1.4 - Prob. 11P Ch. 1.4 - Prob. 12P Ch. 1.4 - Prob. 13P Ch. 1.4 - Prob. 14P Ch. 1.4 - Prob. 15P Ch. 1.4 - Prob. 16P Ch. 1.4 - Prob. 17P Ch. 1.4 - Prob. 18P Ch. 1.4 - Prob. 19P Ch. 1.4 - Prob. 20P Ch. 1.4 - Prob. 21P Ch. 1.4 - 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Program Description: Purpose of the problem is to find the age of the universe.

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Program Description: Purpose of the problem is to find the age of the universe.

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Chapter 1 Solutions