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u' (t) = u(t)- sin(t) + cos(t). a) Find a general solution.

u' (t) = u(t)- sin(t) + cos(t). a) Find a general solution.

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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Page 101 of 614 from Section 2.4.2. i u' (t) = u(t)- sin(t)+cos(t). C Exercise 3.1.6 Consider the linear ODE ✔ 3 Al > Ⓒ Qv 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, U2, . 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 just 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 oneluticel solution to this initial value problem and show that it dogous to zero how >
Transcribed Image Text:Page 101 of 614 from Section 2.4.2. i u' (t) = u(t)- sin(t)+cos(t). C Exercise 3.1.6 Consider the linear ODE ✔ 3 Al > Ⓒ Qv 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, U2, . 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 just 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 oneluticel solution to this initial value problem and show that it dogous to zero how >
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