Explanation Given information: y ' = g ( t , y ) = − ( t + 1 ) y 2 with the initial condition y ( 0 ) = 1 . Formula used: The endpoints ( t 0 , y 0 ) , ( t 1 , y 1 ) , ... , ( t n , y n ) of the line segments approximating the solution of the equation y ' = g ( t , y ) , y ( a ) = y 0 on the interval a ≤ t ≤ b are given by the following formulas where h = b − a n : t 0 = a , y 0 t 1 = t 0 + h , y 1 = y 0 + g ( t 0 , y 0 ) ⋅ h t 2 = t 1 + h , y 2 = y 1 + g ( t 1 , y 1 ) ⋅ h . t n = t n − 1 + h , y n = y n − 1 + g ( t n − 1 , y n − 1 ) ⋅ h Calculation: Here, g ( t , y ) = − ( t + 1 ) y 2 , a = 0 , b = 1 , t 0 = 0 , y 0 = 1 , n = 5 So, h = b − a n = 1 − 0 5 = 1 5 = 0.2 ⇒ h = 0.2 . Now, by using formulas, t 1 = 0 + 0.2 = 0.2 and y 1 = 1 + ( g ( 0 , 1 ) × 0.2 ) = 1 + ( ( − ( 0 + 1 ) × 1 2 ) × 0.2 ) = 1 − 0.2 = 0.8 ⇒ t 1 = 0.2 and y 1 = 0.8 t 2 = 0.2 + 0.2 = 0.4 and y 2 = 0.8 + ( g ( 0.2 , 0.8 ) × 0.2 ) = 0.8 + ( ( − ( 0.2 + 1 ) × 0.8 2 ) × 0.2 ) = 0.8 − 0.1536 = 0.6464 ⇒ t 2 = 0.4 and y 2 = 0.6464 . t 3 = 0.4 + 0.2 = 0.6 and y 3 = 0.6464 + ( g ( 0.4 , 0.6464 ) × 0.2 ) = 0.6464 + ( ( − ( 0.4 + 1 ) × 0.6464 2 ) × 0.2 ) = 0.6464 − 0.1170 = 0.5294 ⇒ t 3 = 0.6 and y 3 = 0.5294 . t 4 = 0.6 + 0.2 = 0.8 and y 4 = 0.5294 + ( g ( 0.6 , 0.5294 ) × 0

Student Solutions Manual for Calculus & Its Applications and Calculus & Its Applications, Brief Version

14th Edition

ISBN: 9780134463230

Author: Larry J. Goldstein, David I Lay, David I. Schneider, Nakhle H. Asmar

Publisher: PEARSON

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Chapter 10.7, Problem 9E

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To calculate: The value of f(1) where f(t) is the solution of y'=−(t+1)y2 with the initial condition y(0)=1 using Euler’s method with number of iterations n=5 and check how accurate is the estimated value by solving the differential equation.

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Chapter 10.7, Problem 8E

Chapter 10.7, Problem 10E

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Chapter 10 Solutions

Student Solutions Manual for Calculus & Its Applications and Calculus & Its Applications, Brief Version

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The value of f ( 1 ) where f ( t ) is the solution of y ' = − ( t + 1 ) y 2 with the initial condition y ( 0 ) = 1 using Euler’s method with number of iterations n = 5 and check how accurate is the estimated value by solving the differential equation.

Solution Summary: The author calculates the value of f(1) using Euler's method with number of iterations and checks how accurate is the estimated value by solving the differential equation.

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To calculate: The value of f(1) where f(t) is the solution of y'=−(t+1)y2 with the initial condition y(0)=1 using Euler’s method with number of iterations n=5 and check how accurate is the estimated value by solving the differential equation.

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Chapter 10 Solutions