Explanation Given: The given differential equation is, y ′ = x − y y ( 0 ) = 0 Formula used: The recursive formulas for Taylor method of order fourth are, x n + 1 = x n + h y n + 1 = y n + h f ( x n , y n ) + h 2 2 f 2 ( x n , y n ) + h 3 3 ! f 3 ( x n , y n ) + h 4 4 ! f 4 ( x n , y n ) Calculation: Consider the differential equation. y ′ = x − y …… ( 1 ) The differential equation is the function of x and y so, y ′ = f ( x , y ) , y ( x 0 ) = y 0 …… ( 2 ) Compare equation ( 1 ) and equation ( 2 ) . f ( x , y ) = x − y x 0 = 0 y o = 0 Differentiate the function f ( x , y ) = x − y with respect to x , f x ( x , y ) = 1 Differentiate the function f ( x , y ) = x − y with respect to x , f y ( x , y ) = − 1 To calculate, f 2 ( x , y ) = f x + f y ( f ( x , y ) ) …… ( 3 ) f 3 ( x , y ) = y ‴ = d f 2 d x …… ( 4 ) f 4 ( x , y ) = d f 3 d x …… ( 5 ) Substitute 1 for f x , − 1 for f y and x − y for f ( x , y ) in equation ( 3 ) . f 2 ( x , y ) = 1 − x + y Differentiate equation ( 4 ) with respect to x . f 3 ( x , y ) = d d x ( 1 − x + y ) = − 1 + d y d x = − 1 + x − y Differentiate equation ( 5 ) with respect to x . f 4 ( x , y ) = d d x ( − 1 + x − y ) = 1 − d y d x = 1 − x + y Consider the recursive formulas for Taylor method of order four

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 3.7, Problem 3E

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The recursive formulas for the Taylor method of order 4.

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Fundamentals of Differential Equations and Boundary Value Problems

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The recursive formulas for the Taylor method of order 4.