Explanation Given: The given differential equation is y ' = x y + y . Concept: The general formulas used in the given problem are, k 1 = h F ( x n , y n ) k 2 = h F ( x n + h 2 , y n + k 1 2 ) k 3 = h F ( x n + h 2 , y n + k 2 2 ) k 4 = h F ( x n + h , y n + k 3 ) And, y n + 1 = y n + 1 6 ( k 1 + 2 k 2 + 2 k 3 + k 4 ) Calculation: The given differential equation can be written as, y ' = F ( x , y ) F ( x , y ) = x y + y The given boundary conditions can be written as, y 0 = f ( x 0 ) x 0 = 0 y 0 = 1 h = 0.1 For n = 0 , Substitute the value of x 0 , y 0 and h in the following equations to get the values of k 1 ,k 2, k 3, k 4 , get the values of k 1 ,k 2, k 3, k 4 , k 1 = h F ( x 0 , y 0 ) = 0.1 × ( ( 0 × 1 ) + ( 1 ) ) = 0.1 × ( 1 ) = 0.1 The value of k 2 is calculated as, k 2 = h F ( x 0 + h 2 , y 0 + k 1 2 ) = 0.1 × F ( 0 + 0.1 2 , 1 + 0.1 2 ) = 0.1 × F ( 0.05 , 1.05 ) = 0.108 The value of k 3 is calculated as, k 3 = h F ( x 0 + h 2 , y 0 + k 2 2 ) = 0.1 × F ( 0 + 0.1 2 , 1 + 0.10772 2 ) = 0.1 × F ( 0.05 , 1.05386 ) = 0.10793 The value of k 4 is calculated as, k 4 = h F ( x 0 + h , y 0 + k 3 ) = 0.1 × F ( 0 + 0.1 , 1 + 0.10793 ) = 0.1 × F ( 0.1 , 1.108 ) = 0.1164 Now substitute the value of k 1 , k 2 , k 3 and k 4 in y 1 . y 1 = y 0 + 1 6 ( k 1 + 2 k 2 + 2 k 3 + k 4 ) = 1 + 1 6 ( 0.1 + 2 ( 0.108 ) + 2 ( 0.10793 ) + ( 0.1164 ) ) = 1 + 1 6 × ( 0.1 + 0.116 + 0.221586 + 0.1164 ) = 1.1079 For n = 1 ,the value of x 1 is calculated as, x 1 = x 0 + h = 0 + 0.1 = 0.1 Substitute the value of x 1 , y 1 and h in the following equations to get the values of k 1 , k 2, k 3, k 4 , k 1 = h F ( x 1 , y 1 ) = 0.1 × ( ( 0.1 ) ( 1.1079 ) + ( 1.1079 ) ) = 0.1 × ( 0.1000 ) = 0.11634 The value of k 2 is calculated as, k 2 = h F ( x 1 + h 2 , y 1 + k 1 2 ) = 0.1 × F ( 0.1 + 0.1 2 , 1.1079 + ( 0.11634 ) 2 ) = 0.1 × F ( 0.15 , 1.6896 ) = 0.15533 The value of k 3 is calculated as, k 3 = h F ( x 1 + h 2 , y 1 + k 2 2 ) = 0.1 × F ( 0.1 + 0.1 2 , 1.1079 + ( 0.15533 ) 2 ) = 0.1 × F ( 0.15 , 1.88454 ) = 0.1655465 The value of k 4 is calculated as, k 4 = h F ( x 1 + h , y 1 + k 3 ) = 0.1 × F ( 0.1 + 0.1 , 1.1079 + ( 0.1655465 ) ) = 0.1 × F ( 0.2 , 1.27345 ) = 0.138316 Now substitute the value of k 1 , k 2 , k 3 and k 4 in y 2 . y 2 = y 1 + 1 6 ( k 1 + 2 k 2 + 2 k 3 + k 4 ) = 1.1079 + 1 6 ( ( 0.11634 ) + 2 ( 0.15533 ) + 2 ( 0.1655465 ) + ( 0.138316 ) ) = 1.2337 For n = 2 , the value of x 2 is calculated as, x 2 = x 1 + h = 0.1 + 0.1 = 0.2 Now Substitute the value of x 2 , y 2 and h in the following equations to get the values of k 1 , k 2, k 3, k 4 , The value of k 1 is calculated as k 1 = h F ( x 2 , y 2 ) = 0.1 × ( ( 0.2 ) ( 1.2337 ) + ( 1.2337 ) ) = 0.1 × ( 1.35746 ) = 0.135746 The value of k 2 is calculated as, k 2 = h F ( x 2 + h 2 , y 2 + k 1 2 ) = 0.1 × F ( 0.2 + 0.1 2 , 1

WEB ASSIGN FOR ZILL'S DIFFERENTIAL EQUAT

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ISBN: 9780357539545

Author: ZILL

Publisher: CENGAGE L

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Chapter 9.2, Problem 10E

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To find: The four decimal approximation of the indicated value using the RK4 method.

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Chapter 9.2, Problem 9E

Chapter 9.2, Problem 11E

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Chapter 9 Solutions

WEB ASSIGN FOR ZILL'S DIFFERENTIAL EQUAT

Ch. 9.1 - In Problems 110 use the improved Eulers method to...Ch. 9.1 - In Problems 1–10 use the improved Euler’s method...Ch. 9.1 - In Problems 1–10 use the improved Euler’s method...Ch. 9.1 - In Problems 1–10 use the improved Euler’s method...Ch. 9.1 - Prob. 5E Ch. 9.1 - Prob. 6E Ch. 9.1 - In Problems 1–10 use the improved Euler’s method...Ch. 9.1 - In Problems 1–10 use the improved Euler’s method...Ch. 9.1 - Prob. 9E Ch. 9.1 - Prob. 10E

Ch. 9.1 - Prob. 11E Ch. 9.1 - Prob. 13E Ch. 9.1 - Prob. 14E Ch. 9.1 - Prob. 15E Ch. 9.1 - Prob. 16E Ch. 9.1 - Consider the initial-value problem y = 2x 3y + 1,...Ch. 9.1 - Prob. 18E Ch. 9.1 - Prob. 19E Ch. 9.1 - Repeat Problem 19 using the improved Euler’s...Ch. 9.1 - Prob. 21E Ch. 9.2 - Use the RK4 method with h = 0.1 to approximate...Ch. 9.2 - Prob. 2E Ch. 9.2 - Prob. 3E Ch. 9.2 - Prob. 4E Ch. 9.2 - Prob. 5E Ch. 9.2 - Prob. 6E Ch. 9.2 - Prob. 7E Ch. 9.2 - Prob. 8E Ch. 9.2 - Prob. 9E Ch. 9.2 - Prob. 10E Ch. 9.2 - In Problems 312 use the RK4 method with h = 0.1 to...Ch. 9.2 - Prob. 12E Ch. 9.2 - Prob. 13E Ch. 9.2 - Consider the initial-value problem y′ = 2y, y(0) =...Ch. 9.2 - Prob. 17E Ch. 9.2 - Consider the initial-value problem y′ = 2x – 3y +...Ch. 9.2 - Prob. 19E Ch. 9.2 - Prob. 20E Ch. 9.3 - Prob. 1E Ch. 9.3 - Prob. 3E Ch. 9.3 - Prob. 4E Ch. 9.3 - Prob. 5E Ch. 9.3 - Prob. 6E Ch. 9.3 - Prob. 7E Ch. 9.3 - Prob. 8E Ch. 9.4 - Use Euler’s method to approximate y(0.2), where...Ch. 9.4 - Prob. 2E Ch. 9.4 - Prob. 3E Ch. 9.4 - Prob. 4E Ch. 9.4 - Prob. 5E Ch. 9.5 - Prob. 1E Ch. 9.5 - Prob. 2E Ch. 9.5 - Prob. 3E Ch. 9.5 - Prob. 4E Ch. 9.5 - Prob. 5E Ch. 9.5 - In Problems 1-18 use Definition 7.1.1 to find ℒ{f...Ch. 9.5 - Prob. 7E Ch. 9.5 - Prob. 8E Ch. 9.5 - Prob. 9E Ch. 9.5 - Prob. 10E Ch. 9.5 - Prob. 11E Ch. 9.5 - The electrostatic potential u between two...Ch. 9.5 - Consider the boundary-value problem y″ + xy = 0,...Ch. 9 - Prob. 1RE Ch. 9 - Prob. 2RE Ch. 9 - In Problems 1–4 construct a table comparing the...Ch. 9 - In Problems 1–4 construct a table comparing the...Ch. 9 - Prob. 5RE Ch. 9 - Prob. 6RE Ch. 9 - Prob. 7RE Ch. 9 - Prob. 8RE

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The four decimal approximation of the indicated value using the RK4 method.

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