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Consider the initial value problem below to answer to following. a) Find the approximations to y(0.4) and y(0.8) using Euler's method with time steps of At = 0.4, 0.2, 0.1, and 0.05. b) Using the exact solution given, compute the errors in the Euler approximations at t= 0.4 and t= 0.8. c) Which time step results in the more accurate approximation? Explain your observations. d) In general, how does halving the time step affect the error at t = 0.4 and t= 0.8? y'(t) = 6t + 1, y(0) = 0, y(t) = 3r² +t

Consider the initial value problem below to answer to following. a) Find the approximations to y(0.4) and y(0.8) using Euler's method with time steps of At = 0.4, 0.2, 0.1, and 0.05. b) Using the exact solution given, compute the errors in the Euler approximations at t= 0.4 and t= 0.8. c) Which time step results in the more accurate approximation? Explain your observations. d) In general, how does halving the time step affect the error at t = 0.4 and t= 0.8? y'(t) = 6t + 1, y(0) = 0, y(t) = 3r² +t

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please make numbers easy to read. Problem is a few parts, A-D.

Consider the initial value problem below to answer to following. a) Find the approximations to y(0.4) and y(0.8) using Euler's method with time steps of At = 0.4, 0.2, 0.1, and 0.05. b) Using the exact solution given, compute the errors in the Euler approximations at t=0.4 and t= 0.8. cj Which time step results in the more accurate approximation? Explain your observations. d) In general, how does halving the time step affect the error at t= 0.4 and t= 0.8? y'(t) = 6t + 1, y(0) = 0, y(t) = 3r² +t a) Complete the table below. At Approximation to y(0.4) Approximation to y(0.8) 0.4 0.2 0.1 0.05 (Round to tive decimal places as needed.) b) Complete the table below. At Errors for y(0.4) Errors for y(0.8) 0.4 0.2 0.1 0.05 (Round five decimal places as needed.) c) Which time step results in the more accurate approximation? O A. Time step At= 0.4 results in the more accurate approximation because larger time steps generally produce more accurate results. O B. Time step At= 0.4 results in the more accurate approximation because larger time steps generally produce smaller numbers. O C. Time step At=0.05 results in the more accurate approximation because smaller time steps generally produce more accurate results. O D. Time step At = 0.05 results in the more accurate approximation because smaller time steps generally produce smaller numbers. d) How did halving the time step affect the error at t= 0.4 and t= 0.8? O A. Halving the time steps results in approximately doubling the error. B. Halving the time steps results in reducing the error by approximately one tenth. OC. Halving the time steps results in approximately halving the error. O D. Halving the time steps results in reducing the error by approximately one fourth.
Transcribed Image Text:Consider the initial value problem below to answer to following. a) Find the approximations to y(0.4) and y(0.8) using Euler's method with time steps of At = 0.4, 0.2, 0.1, and 0.05. b) Using the exact solution given, compute the errors in the Euler approximations at t=0.4 and t= 0.8. cj Which time step results in the more accurate approximation? Explain your observations. d) In general, how does halving the time step affect the error at t= 0.4 and t= 0.8? y'(t) = 6t + 1, y(0) = 0, y(t) = 3r² +t a) Complete the table below. At Approximation to y(0.4) Approximation to y(0.8) 0.4 0.2 0.1 0.05 (Round to tive decimal places as needed.) b) Complete the table below. At Errors for y(0.4) Errors for y(0.8) 0.4 0.2 0.1 0.05 (Round five decimal places as needed.) c) Which time step results in the more accurate approximation? O A. Time step At= 0.4 results in the more accurate approximation because larger time steps generally produce more accurate results. O B. Time step At= 0.4 results in the more accurate approximation because larger time steps generally produce smaller numbers. O C. Time step At=0.05 results in the more accurate approximation because smaller time steps generally produce more accurate results. O D. Time step At = 0.05 results in the more accurate approximation because smaller time steps generally produce smaller numbers. d) How did halving the time step affect the error at t= 0.4 and t= 0.8? O A. Halving the time steps results in approximately doubling the error. B. Halving the time steps results in reducing the error by approximately one tenth. OC. Halving the time steps results in approximately halving the error. O D. Halving the time steps results in reducing the error by approximately one fourth.
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