The final exam is scheduled for 4:00-5:50 pm, Monday, Dec. 15,
2003. Students are strongly encouraged to work together on
homework assignments. Late homework will be accepted only
with the advanced permission of the instructor.
Work exercises 1.3cdi, 1.5a, 1.6, 1.8, 1.9, 1.10 on pp. 49-50.
For each of 1.5a, 1.8, 1.9, 1.10, generate plots of the (numerically
computed) "true solution" together with plots of your asymptotic
approximation.
Based on your plots, guess whether the asymptotic
approximations converge pointwise or converge uniformly for t > 0.
For extra credit, provide ***mathematical proof*** of your guesses.
I highly recommend that you use mathematical software like MAPLE,
MATHCAD, or MATHEMATICA, or a calculator that does symbolic computations,
to evaluate any nontrivial integrals, derivatives, or solutions to
algebraic systems. If you do so, please ***insert comments in your
homework*** to explain what you have done.
Homework Assignment 6: Due Monday, Nov. 10.
Work exercises 3.2b,3.3,3.4,3.7c on pp. 69-70.
For each exercise except 3.3, hand in plots of the
uniform approximations.
For exercise 3.3, hand in a plot of the exact solution.
Work exercises 3.1ac, 3.2a on pp. 130-131. Don't rely too much
on formulas. For 3.1c, find a sequence of admissible functions
y_n(x) for which lim J(y_n) = 1/2 (1/2 is an upper bound for J(y)),
but the pointwise limit of y_n(x) is not admissible.
Work exercises 3.4, 3.6, 3.7 on p. 131-132.
Required for math majors who have had analysis and an upper-level
coarse in linear algebra: Exercises 2.1abc, 2.2, and 2.4 on p. 121.
Homework Assignment 9: Due Wednesday, Dec. 10.
Work exercises 4.2a, 4.4a, 4.5b, 4.12a, 4.17 on pp. 145-148.
Find the Euler equation and the boundary conditions which arise
in the minimization of J(y) = int_0^1 [(y''(x)^2 - f(x)y(x)] dx,
subject to y'(0) = 5, y(1) = 8. Which boundary conditions are essential?
Which are natural?