(a) Explanation Calculation: The given initial value problem is: y ' = 1 − y 3 , y ( 0 ) = 0 A differential equation of the form y ' + p ( t ) y + q ( t ) = c is said to be initial value problem if it is associated with some initial value such as y ( t 0 ) = y 0 . If y = ϕ ( t ) be a differentiable function satisfies initial value problem, then the function is obtained by integrating y ' = f ( t , y ) from the initial point t = 0 to any arbitrary point t. ϕ ( t ) = ∫ 0 t f ( s , ϕ ( s ) ) d s The general form of the method of successive approximation is as follows: ϕ n + 1 ( t ) = ∫ 0 t f ( s , ϕ n ( s ) ) d s Here, each term in the sequence ϕ n satisfies the initial condition. Let y = ϕ ( t ) be a differential function satisfies initial value problem, then the corresponding integral equation is as follows: ϕ n ( t ) = ∫ 0 t [ 1 − ϕ n − 1 ( s ) 3 ] d s Now, the initial approximation is as follows: ϕ 0 ( t ) = 0 It shows that, ϕ 1 ( t ) = ∫ 0 t [ 1 − ϕ 0 ( s ) 3 ] d s = ∫ 0 t ( 1 − 0 ) d s = [ s ] 0 t = t Similarly calculate the second value of the approximation is: ϕ 2 ( t ) = ∫ 0 t [ 1 − ϕ 1 ( s ) 3 (b) To determine To plot: The function ϕ 1 ( t ) , ... , ϕ 3 ( t ) and observe whether the iterates appear to be converging.

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Chapter 2.8, Problem 10P

(a)

To determine

The value of the ϕ1(t),...,ϕ3(t).

(b)

To determine

To plot: The function ϕ1(t),...,ϕ3(t) and observe whether the iterates appear to be converging.

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Chapter 2.8, Problem 9P

Chapter 2.8, Problem 11P

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Chapter 2 Solutions

Elementary Differential Equations

Ch. 2.1 - In each of Problems 1 through 12: Draw a direction...Ch. 2.1 - In each of Problems 1 through 12: Draw a direction...Ch. 2.1 - Prob. 3P Ch. 2.1 - Prob. 4P Ch. 2.1 - Prob. 5P Ch. 2.1 - Prob. 6P Ch. 2.1 - In each of Problems 1 through 12: Draw a direction...Ch. 2.1 - Prob. 8P Ch. 2.1 - Prob. 9P Ch. 2.1 - Prob. 10P

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The value of the ϕ 1 ( t ) , ... , ϕ 3 ( t ) .

The value of the ϕ1(t),...,ϕ3(t).

To plot: The function ϕ1(t),...,ϕ3(t) and observe whether the iterates appear to be converging.

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Chapter 2 Solutions