a . Explanation Given: The Differential equation is y ′ ( t ) = 4 − y , y ( 0 ) = 3 and Δ t = 0.2 . Calculation: The Differential equation is y ′ ( t ) = 4 − y with the time steps Δ t = 0.2 . The set of grid points are t 0 = 0.0 , t 1 = 0.2 and t 2 = 0.4 Take f ( t , y ) = 4 − y and let u k be the Euler’s approximation y ( t k ) . Consider u 0 = y ( 0 ) = 3 and t 0 = 0.0 . The Euler’s method to find the value of u k is, u k + 1 = u k + f ( t k , u k ) Δ t ( ∵ k = 0 , 1 , 2 , 3.... ) (1) Take t k be the time steps as given in the table. Consider t 0 = 0 and substitute the value of t 0 = 0 in the equation (1) and find the value of u k . Consider the time step size Δ t = 0.2 and k = 0 b . To determine To Compute: The error in the Euler approximation at t = 0.2 and t = 0.4 using the exact solution. c . To determine To Explain: The time step which give more accurate approximation. d . To determine The error when the time step is halved.

2nd Edition

ISBN: 9781323178522

Author: Briggs

Publisher: PEARSON

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Differential Equations

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Chapter D1.2, Problem 31E

To determine

To approximate: The value of y(0.2) and y(0.4) using Euler’s method with the given time steps

To determine

To Compute: The error in the Euler approximation at t=0.2 and t=0.4 using the exact solution.

To determine

To Explain: The time step which give more accurate approximation.

To determine

The error when the time step is halved.

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Chapter D1.2, Problem 30E

Chapter D1.2, Problem 32E

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To approximate: The value of y ( 0.2 ) and y ( 0.4 ) using Euler’s method with the given time steps

Solution Summary: The approximated value of the differential equation is given in table shown in below.

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To approximate: The value of y(0.2) and y(0.4) using Euler’s method with the given time steps

To Compute: The error in the Euler approximation at t=0.2 and t=0.4 using the exact solution.

To Explain: The time step which give more accurate approximation.

The error when the time step is halved.

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