To determine The value of y ( 2 ) using Euler’s method for the function y ′ = x y with x 0 = 1 and d x = 0.2 and also to find the exact value of y ( 2 ) for comparison, to sketch the slope field and a solution curve passing through the point ( 1 , − 1 ) . Explanation Formula used: The first approximation of Euler’s method is given by y 1 = L ( x 1 ) = y 0 + f ( x 0 , y 0 ) d x . The second approximation of Euler’s method is given by y 2 = y 1 + f ( x 1 , y 1 ) d x . The third approximation of Euler’s method is given by y 3 = y 2 + f ( x 2 , y 2 ) d x . Since y ( 1 ) = − 1 , x 0 = 1 and y 0 = − 1 . Evaluate the first approximation as follows. y 1 = y 0 + f ( x 0 , y 0 ) d x = y 0 + x 0 y 0 d x = − 1 + ( 1 ) ( − 1 ) ( 0.2 ) ( ∵ x 0 = 1 , y 0 = − 1 and d x = 0.2 ) = − 1 − 0.2 = − 1.2 Thus, the first approximation is –1.2. Evaluate the second approximation as follows. y 2 = y 1 + f ( x 1 , y 1 ) d x = y 1 + x 1 y 1 d x = − 1.2 + ( 1.2 ) ( − 1.2 ) ( 0.2 ) ( ∵ x 1 = x 0 + d x , y 1 = − 1.2 and d x = 0.2 ) = − 1.488 Thus, the second approximation is − 1.488 . Evaluate the third approximation as follows. y 3 = y 2 + f ( x 2 , y 2 ) d x = y 2 + x 2 y 2 d x = − 1.488 + ( 1.4 ) ( − 1.488 ) ( 0.2 ) ( ∵ x 2 = x 1 + d x , y 2 = − 1.488 and d x = 0.2 ) = − 1.9046 Thus, the third approximation is − 1.9046 . Evaluate the fourth approximation as follows. y 4 = y 3 + f ( x 3 , y 3 ) d x = y 3 + x 3 y 3 d x = − 1.9046 + ( 1.6 ) ( − 1.9046 ) ( 0.2 ) ( ∵ x 3 = x 2 + d x , y 3 = − 1.9046 and d x = 0.2 ) = − 2

Student Solutions Manual For Thomas' Calculus Format: Paperback

14th Edition

ISBN: 9780134439334

Author: Hass, Joel R.^heil, Christopher D.^weir, Maurice D.^heil, Christopher

Publisher: PEARSON

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Chapter 9, Problem 37PE

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The value of y(2) using Euler’s method for the function y′=xy with x0=1 and dx=0.2 and also to find the exact value of y(2) for comparison, to sketch the slope field and a solution curve passing through the point (1,−1).

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Chapter 9, Problem 36PE

Chapter 9, Problem 38PE

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Student Solutions Manual For Thomas' Calculus Format: Paperback

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