Exercise 3.1.6 Consider the linear ODE u' (t) = u(t)- sin(t) + cos(t). (a) Find a general solution. (b) Find the solution with initial condition u(0) = 0. (c) Sketch a direction field on the range 0 ≤ t ≤ 10,-5 ≤u≤ 5, and superimpose the solution with u(0) = 0 on this direction field. (d) Apply Euler's method with step sizes h= 1,0.1,0.01, 0.001 with initial condition u(0) = 0 to estimate u(10). Explain the poor estimates for u(10) in light of the direction field from (c) and general solution from (a). Hint: what happens if Euler's method ever strays from the analytical solution curve? It might be helpful to plot the Euler iterates uo, u1, U2, .. for the case h = 0.001.
Exercise 3.1.6 Consider the linear ODE u' (t) = u(t)- sin(t) + cos(t). (a) Find a general solution. (b) Find the solution with initial condition u(0) = 0. (c) Sketch a direction field on the range 0 ≤ t ≤ 10,-5 ≤u≤ 5, and superimpose the solution with u(0) = 0 on this direction field. (d) Apply Euler's method with step sizes h= 1,0.1,0.01, 0.001 with initial condition u(0) = 0 to estimate u(10). Explain the poor estimates for u(10) in light of the direction field from (c) and general solution from (a). Hint: what happens if Euler's method ever strays from the analytical solution curve? It might be helpful to plot the Euler iterates uo, u1, U2, .. for the case h = 0.001.
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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Do all parts
Transcribed Image Text:Page 101 of 614
from Section 2.4.2.
D
Exercise 3.1.6 Consider the linear ODE
u' (t) = u(t)- sin(t) + cos(t).
3
0
☹
Q Search
(a) Find a general solution.
(b) Find the solution with initial condition u(0) = 0.
(c) Sketch a direction field on the range 0 ≤ t ≤ 10,-5 ≤u≤ 5, and superimpose the solution
with u(0) = 0 on this direction field.
(d) Apply Euler's method with step sizes h= 1,0.1,0.01, 0.001 with initial condition u(0) = 0
to estimate u(10). Explain the poor estimates for u(10) in light of the direction field from
(c) and general solution from (a). Hint: what happens if Euler's method ever strays from
the analytical solution curve? It might be helpful to plot the Euler iterates uo, u₁, 2, ..
for the case h = 0.001.
Exercise 3.1.7 This problem illustrates that if the step size is too large, Euler's method isn't
ust inaccurate-it may actually blow up, even if the true solution to the ODE decays. This
should also be apparent in part (f) of Exercise 3.1.4.
Consider the differential equation u' (t) = -10u(t) with u(0) = 1.
(n) Find the onolutical colution to this initial voluo problem and show that it dogove to zoro
G
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