Explanation Consider the given differential equations with the given initial conditions. y ′ = y − y 2 , y ( 0 ) = 0.5 The Euler’s method formula is given as, y n + 1 = y n + h f ( x n , y n ) , where x n = x 0 + n h , n = 0 , 1 , 2 , … . Given differential equation is written in terms of Euler’s formula as follows. y n + 1 = y n + h ( y n − y n 2 ) The value of y 1 is evaluated for h = 0.1 , x 0 = 0 , y 0 = 0.5 and n = 0 as follows. y 1 = y 0 + 0.1 ( y 0 − y 0 2 ) = 0.5 + 0.1 ( 0.5 − ( 0.5 ) 2 ) = 0.525 This is the estimate value of y ( 0.1 ) = 0.525 . Now, when the smaller step size h = 0.05 is used, it takes two steps to reach x = 0.1 . y 1 = y 0 + 0.05 ( y 0 − y 0 2 ) = 0.5125 y 2 = y 1 + 0.05 ( y 1 − y 1 2 ) = 0

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ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 2.6, Problem 10E

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To solve: The given differential equation y′=y−y2 with numerical solver and Euler’s method to obtain a four decimal approximation of y(0.5) by initially using h=0.1 and then h=0.05.

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Chapter 2.6, Problem 9E

Chapter 2.7, Problem 1E

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The given differential equation y ′ = y − y 2 with numerical solver and Euler’s method to obtain a four decimal approximation of y ( 0.5 ) by initially using h = 0.1 and then h = 0.05 .

Solution Summary: The author explains how to solve the given differential equation with numerical solver and Euler's method to obtain a four decimal approximation of y(0.5).

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To solve: The given differential equation y′=y−y2 with numerical solver and Euler’s method to obtain a four decimal approximation of y(0.5) by initially using h=0.1 and then h=0.05.

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Chapter 2 Solutions