Explanation Given: The given differential equation is, y = x 2 + 2 y 2 , y ( 0 ) = − 3 , h = 0.1 , y ( 1 ) ... ( 1 ) Approach: The fourth order Runge-Kutta method for approximating the solution is given by, y n + 1 = y n + 1 6 ( k 1 + 2 k 2 + 2 k 3 + k 4 ) ... ( 2 ) Here, k 1 = h f ( x n , y n ) , k 2 = h f ( x n + 1 2 h , y n + 1 2 k 1 ) , k 3 = h f ( x n + 1 2 h , y n + 1 2 k 2 ) And k 4 = h f ( x n + 1 , y n + k 3 ) Calculation: From given data, Here, h = 0.1 So, k 1 = ( 0.1 ) f ( x n , y n ) , = ( 0.1 ) ( x n 2 , 2 y n 2 ) Similarly, k 2 = ( 0.1 ) ( x n + 0.05 , y n + ( 0.5 ) k 1 ) = ( 0.1 ) ( ( x n + 0.05 ) 2 , 2 ( y n + 0.5 k 1 ) 2 ) Similarly, k 3 = ( 0.1 ) f ( x n + 0.5 , y n + 0.5 k 2 ) = ( 0.1 ) f ( ( x n + 0.5 ) 2 , 2 ( y n + 0.5 k 2 ) 2 ) And, k 4 = ( 0.1 ) ( x n + 1 , y n + k 3 ) = ( 0.1 ) ( ( x n + 1 ) 2 , 2 ( y n + k 3 ) 2 ) Substitute 0 for n in above equations. Here, k 1 = ( 0.1 ) ( ( 0 ) n 2 , 2 ( − 3 ) n 2 ) = 1.8 Similarly, k 2 = ( 0.1 ) ( ( 0 + 0.05 ) 2 , 2 ( − 3 + 0.5 ( 1.8 ) ) 2 ) = 0.8822 Similarly, k 3 = ( 0.1 ) ( ( 0 + 0.5 ) 2 , 2 ( − 3 + 0.5 ( 0.8822 ) ) 2 ) = 1.3098 Similarly, k 4 = ( 0.1 ) ( ( 0

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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Chapter 1.12, Problem 62AP

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The solution to the given initial- value problem at the specified point and compare the answer to that determined by Euler’s method.

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Chapter 1.12, Problem 61AP

Chapter 1.12, Problem 63AP

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Chapter 1 Solutions

Differential Equations and Linear Algebra (4th Edition)

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The solution to the given initial- value problem at the specified point and compare the answer to that determined by Euler’s method.

Solution Summary: The author explains the fourth order Runge-Kutta method for approximating the solution to the given initial-value problem at the specified point.

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The solution to the given initial- value problem at the specified point and compare the answer to that determined by Euler’s method.

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