Explanation Given: The initial value problem solution is, y ' = y + x y y ( 0 ) = 1 Step size = 0.1 Calculation: Write the general formula for Euler’s approximation method, y ' = F ( x , y ) y ( x 0 ) = y 0 y n = y n − 1 + h F ( x n − 1 , y n − 1 ) x n = x n − 1 + h Here h is the step size and n = 1 , 2 , 3 , 4 , ... By comparing the above mentioned general formula with the given equation, we have the following, F ( x , y ) = y + x y x 0 = 0 y 0 = 1 h = 0.1 Write the formula for y 1 , y 1 = y 0 + h F ( x 0 , y 0 ) Substitute 1 for y 0 , 0 for x 0 and 0.1 for h in above equation, y 1 = 1 + 0.1 F ( 0 , 1 ) F ( 0 , 1 ) = y + x y = 1 + ( 0 ) ( 1 ) = 1 Substitute 1 for F ( 0 , 1 ) in the equation y 1 = 1 + 0.1 F ( 0 , 1 ) , y 1 = 1 + 0.1 ( 1 ) = 1 + 0.1 = 1.1 Write the formula for x 1 . x 1 = x 0 + h Substitute 0 for x 0 and 0.1 for h in above equation, x 1 = 0 + 0.1 = 0.1 Thus the values of x 1 , y 1 are given below, x 1 = 0.1 and y 1 = 1.1 Similarly write the formula for y 2 . y 2 = y 1 + h F ( x 1 , y 1 ) Substitute 1.1 for y 1 , 0.1 for x 1 , 0.1 for h in above equation, y 2 = 1.1 + 0.1 F ( 0.1 , 1.1 ) F ( 0.1 , 1.1 ) = y + x y = 1.1 + ( 0.1 ) ( 1.1 ) = 1.1 + 0.11 = 1.21 Substitute 1.21 for F ( 0.1 , 1.1 ) in the equation y 2 = 1.1 + 0.1 F ( 0.1 , 1.1 ) , y 2 = 1.1 + 0.1 ( 1.21 ) = 1.1 + 0.121 = 1.221 Write the formula for x 2 . x 2 = x 1 + h Substitute 0.1 for x 1 and 0.1 for h in above equation, x 2 = 0.1 + 0.1 = 0.2 Thus the values of x 2 , y 2 are given below, x 2 = 0.2 and y 2 = 1.221 Similarly write the formula for y 3 . y 3 = y 2 + h F ( x 2 , y 2 ) Substitute 1.221 for y 2 , 0.2 for x 2 , 0.2 for h in above equation, y 3 = 1.221 + 0.1 F ( 0.2 , 1.221 ) F ( 0.2 , 1.221 ) = y + x y = 1.221 + ( 0.2 ) ( 1.221 ) = 1

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ISBN: 9781285741550

Author: James Stewart

Publisher: Cengage Learning

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Chapter 9.2, Problem 23E

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To Estimate: The y(0.5) using Euler’s method with the step size of 0.1.

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Chapter 9.2, Problem 22E

Chapter 9.2, Problem 24E

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To Estimate: The y ( 0.5 ) using Euler’s method with the step size of 0.1 .

Solution Summary: The author calculates the value of y(0.5) using Euler's approximation method.

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To Estimate: The y(0.5) using Euler’s method with the step size of 0.1.

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