Math 293, Homework 6, due Wed. October 9, 1996
1) Make up a word problem, set up the equations, and solve
the equations using MATLAB and ODE23, plot your
solution. The equations should be a pair of
coupled first order differential equations.
Sentences like "assume x changes at a rate proportional
to y squared with a proportionality constant of 6" do
NOT count really as word problems!!
Try to think of a model of some physical or social
situation where you think the model is kind of
reasonable.
You may also check your work using pplane or using
Bob Terrel's Applet 'de'. For help with ODE23, for a copy
of pplane, or for info on 'de', consult the course
WWW page.
Hand in: words, ODEs and ICs, MATLAB work, MATLAB output,
your interpretion of the result.
2) You should be able to do 4.1.1-20 and 4.1.26
Hand in 4.21a.
Also do 4.21a) by the following method: find the
derivative (with respect to time) of
x^2 + y^2 assuming only that x and y solve the given system,
i.e., without using the solution to the ODEs for x(t) and y(t).
Show how this result implies that the trajectories must be
circles. Moral: sometimes you can find the phase plane
trajectory without finding the time history of the motion.
*) You should be able to do 5.1.1-6 and 5.1.11-20
*) For A = [ 2 3 ] a two by two matrix, and [b] = [1], a 2x1 col vector,
[ 1 4 ] [5]
a) Find Ab,
b) the transpose of A,
c) the diagonal part of A,
d) write the equation Ax=b in long hand (as scalar equations),
e) solve Ax=b for x.
f) check your solution.
g) Calculate AC where C = [ 2 0 ]. Note that this just multiplies the
[ 0 3 ] columns of A by 2 and 3.
3) For A = 1 2 3 4 5 (5x5 matrix) and b = 1 (5x1 col vector)
5 4 3 2 1 2
1 2 0 0 0 3
5 5 4 4 4 4
3 0 3 0 3 5
repeat steps a-f from problem 2. You may use MATLAB if you like.
(See WWW page for a script file with a list of useful matrix matlab
commands).
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A few people asked about what to hand in, so let me clarify.
A) (*) means you should learn it, but don't spend time doing polished
work to hand in. In other words, you get no direct credit for handing in
any of these problems. Of course, we imagine you get indirect credit
(in knowledge, for other homework problems, in exams, in other courses,
in sense of self confidence, etc ) by gaining competance at them, but
we do not want you to spend time on the written communication aspects
of work that you want others to be able to read and understand.
B) (1), (2), or (3) means you should write up in a nice clear way,
following the homework guidelines (which you can review on the WWW page).
C) Problem (3) makes reference to problem (4) (or (2) depending on
which version you read). This is really a reference to the
previous (*) problem.
Sorry for the confusion.