Exercise 3.2.3 Apply the improved Euler's method to the ODE u' (t) = u(t)-t+1 with u(0) = 0, to estimate u(1) using step sizes h= 1,0.1, and 0.01. Then find the analytical solution and compute the error for each step size. Why does this make perfect sense? Exercise 3.2.4 (Compare the results here to Exercise 3.1.5.) Apply the improved Euler's method to the ODE u' (t) = u²(t) with u(0) = 1 to estimate u(2) using step sizes h = 1,0.1,0.01, and 0.001. Explain what's going on. Hint: compute the analytical solution using separation of variables. Then recall Definition 2.4.1 and the notion of the maximum domain of a solution from Section 2.4.2. Exercise 3.2.5 (Compare the results here to Exercise 3.1.6.) Consider the linear ODE u' (t) = u(t) - sin(t) + cos(t).
Exercise 3.2.3 Apply the improved Euler's method to the ODE u' (t) = u(t)-t+1 with u(0) = 0, to estimate u(1) using step sizes h= 1,0.1, and 0.01. Then find the analytical solution and compute the error for each step size. Why does this make perfect sense? Exercise 3.2.4 (Compare the results here to Exercise 3.1.5.) Apply the improved Euler's method to the ODE u' (t) = u²(t) with u(0) = 1 to estimate u(2) using step sizes h = 1,0.1,0.01, and 0.001. Explain what's going on. Hint: compute the analytical solution using separation of variables. Then recall Definition 2.4.1 and the notion of the maximum domain of a solution from Section 2.4.2. Exercise 3.2.5 (Compare the results here to Exercise 3.1.6.) Consider the linear ODE u' (t) = u(t) - sin(t) + cos(t).
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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Transcribed Image Text:Exercise 3.2.3 Apply the improved Euler's method to the ODE u' (t) = u(t)-t+1 with
u(0) = 0, to estimate u(1) using step sizes h= 1,0.1, and 0.01. Then find the analytical solution
and compute the error for each step size. Why does this make perfect sense?
Exercise 3.2.4 (Compare the results here to Exercise 3.1.5.) Apply the improved Euler's
method to the ODE u' (t) = u²(t) with u(0) = 1 to estimate u(2) using step sizes h = 1,0.1,0.01,
and 0.001. Explain what's going on. Hint: compute the analytical solution using separation of
variables. Then recall Definition 2.4.1 and the notion of the maximum domain of a solution
from Section 2.4.2.
Exercise 3.2.5 (Compare the results here to Exercise 3.1.6.) Consider the linear ODE
u' (t) = u(t)- sin(t) + cos(t).
tv
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