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Consider the initial value problem (a) Using definite

Fundamentals of Differential Equations | 8th Edition | ISBN: 9780321747730 | Authors: R. Kent Nagle, Edward B. Saff, Arthur David Snider ISBN: 9780321747730 43

Solution for problem 27E Chapter 2.3

Fundamentals of Differential Equations | 8th Edition

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Fundamentals of Differential Equations | 8th Edition | ISBN: 9780321747730 | Authors: R. Kent Nagle, Edward B. Saff, Arthur David Snider

Fundamentals of Differential Equations | 8th Edition

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Problem 27E

Consider the initial value problem

(a) Using definite integration, show that the integrating factor for the differential equation can be written as

and that the solution to the initial value problem is

(b) Obtain an approximation to the solution at x = 1 by using numerical integration (such as Simpson’s rule, Appendix C) in a nested loop to estimate values of µ(x) and, thereby, the value of

[Hint: First, use Simpson’s rule to approximate µ (x) at x = 0.1, 0.2,…, 1. Then use these values and apply Simpson’s rule again to approximate

(c) Use Euler’s method (Section 1.4) to approximate the solution at x = 1, with step sizes h = 0.1 and 0.05.

[A direct comparison of the merits of the two numerical schemes in parts (b) and (c) is very complicated, since it should take into account the number of functional evaluations in each algorithm as well as the inherent accuracies.]

Step-by-Step Solution:

Step 1</p>

In this problem we have to  that the integrating factor for the differential equation can be written as

Step 2</p>

Method of solving linear Equations

        

Now, we have to calculate the integrating factor

So, multiplying the standard equation by

        

        

Last step we have to integrate the last equation with respect to y and divide by

Step 3</p>

In the Question it is given that initial value problem

The given is of the form of

Now, we have to calculate the integration factor

        

If we let t be the variable then the upper limit be and dx change to dt

                

Step 4</p>

So, multiplying the standard equation by

        

        

        

Last step we have to integrate the last equation from limit  and divide by

        

Step 5 of 8

Chapter 2.3, Problem 27E is Solved
Step 6 of 8

Textbook: Fundamentals of Differential Equations
Edition: 8
Author: R. Kent Nagle, Edward B. Saff, Arthur David Snider
ISBN: 9780321747730

The full step-by-step solution to problem: 27E from chapter: 2.3 was answered by , our top Calculus solution expert on 07/11/17, 04:37AM. Since the solution to 27E from 2.3 chapter was answered, more than 231 students have viewed the full step-by-step answer. This textbook survival guide was created for the textbook: Fundamentals of Differential Equations , edition: 8. Fundamentals of Differential Equations was written by and is associated to the ISBN: 9780321747730. The answer to “Consider the initial value problem (a) Using definite integration, show that the integrating factor for the differential equation can be written as and that the solution to the initial value problem is (b) Obtain an approximation to the solution at x = 1 by using numerical integration (such as Simpson’s rule, Appendix C) in a nested loop to estimate values of µ(x) and, thereby, the value of [Hint: First, use Simpson’s rule to approximate µ (x) at x = 0.1, 0.2,…, 1. Then use these values and apply Simpson’s rule again to approximate (c) Use Euler’s method (Section 1.4) to approximate the solution at x = 1, with step sizes h = 0.1 and 0.05.[A direct comparison of the merits of the two numerical schemes in parts (b) and (c) is very complicated, since it should take into account the number of functional evaluations in each algorithm as well as the inherent accuracies.]” is broken down into a number of easy to follow steps, and 153 words. This full solution covers the following key subjects: approximate, rule, simpson, use, solution. This expansive textbook survival guide covers 67 chapters, and 2118 solutions.

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