Explanation Given: The given equation is, y ′ = 4 y − 1 The step size and initial condition is given as h = 0.05 and y ( 0 ) = 1 . Formula used: The formula for the Euler modified method is y n + 1 = y n + h 2 [ f ( x n , y n ) + f ( x n + 1 , y n + 1 * ) ] Where, the step size is h and y n + 1 * = y n + h f ( x n , y n ) . Calculation: Consider the given differential equation. y ′ = 4 y − 1 Consider the expression for y n + 1 , y n + 1 = y n + h 2 ( 4 y n - 1 + 4 y n * - 1 ) ........(1) Substitute 0 for n , 0.05 for h and 1 for y 0 in equation (1). y 1 = 1 + 0.025 × ( 4 ( 1 ) − 1 + 4 ( 1.15 ) − 1 ) = 1.165 Substitute 1 for n , 0.05 for h and 1.165 for y 1 in equation (1). y 2 = 1.165 + 0.025 × ( 4 ( 1.165 ) − 1 + 4 [ 1.165 + 0.05 ( 4 × 1.165 − 1 ) ] − 1 ) = 1.3663 Substitute 2 for n , 0.05 for h and 1.3663 for y 2 in equation (1). y 3 = 1.3663 + 0.025 × ( 4 ( 1.3663 ) − 1 + 4 [ 1.3663 + 0.05 ( 4 × 1.3663 − 1 ) ] − 1 ) = 1.6119 Substitute 3 for n , 0.05 for h and 1.6119 for y 3 in equation (1). y 4 = 1.6119 + 0.025 × ( 4 ( 1.6119 ) − 1 + 4 [ 1.6119 + 0.05 ( 4 × 1.6119 − 1 ) ] − 1 ) = 1.9115 Substitute 4 for n , 0.05 for h and 1.9115 for y 4 in equation (1). y 5 = 1.9115 + 0.025 × ( 4 ( 1.9115 ) − 1 + 4 [ 1.9115 + 0.05 ( 4 × 1.9115 − 1 ) ] − 1 ) = 2.2770 Substitute 5 for n , 0.05 for h and 2.2770 for y 5 in equation (1). y 5 = 2.2770 + 0.025 × ( 4 ( 2.2770 ) − 1 + 4 [ 2.2770 + 0.05 ( 4 × 2.2770 − 1 ) ] − 1 ) = 2.7230 Substitute 6 for n , 0.05 for h and 2.7230 for y 6 in equation (1). y 5 = 2.7230 + 0.025 × ( 4 ( 2.7230 ) − 1 + 4 [ 2.7230 + 0.05 ( 4 × 2.7230 − 1 ) ] − 1 ) = 3

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ISBN: 9780321990167

Author: ANNIN

Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)

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Chapter 1.10, Problem 6P

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The value of y(0.5) using Euler’s modified method.

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The value of y ( 0.5 ) using Euler’s modified method.

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

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To find:

The value of y(0.5) using Euler’s modified method.

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The value of y ( 0.5 ) using Euler’s modified method.

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

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To find: The value of y(0.5) using Euler’s modified method.

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Chapter 1 Solutions

To find:

The value of y(0.5) using Euler’s modified method.