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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) = 3t² +1 +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) = 3t² +1 +t

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 10CR
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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) = 3t² + 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 five decimal places as needed.)
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. 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) = 3t² + 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 five decimal places as needed.)
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