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Syllabus
Homework assignments
Using your computer
About the exams
Notes, links and slides
Mo We Fr
19 21 Jan
24 26 28
31 2 4 Feb
7 9 11
14 16 18
21 23 25
28 2 4 Mar
7 9 11
SPRING BREAK
21 23 25
28 30 1
4 6 8 Apr
11 13 15
18 20 22
25 27 29
2 4 6 May
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Homework will be assigned every monday in
lecture, and later posted on this page. Homework must be
returned in lecture on the following monday. It should be
neatly written; separate sheets
must be stapled (no paper clips, etc.); your name
must appear on each sheet. Homework that does not meet
these standards will not be graded.
Due Monday May 2.
Our last topic is about solving inhomogeneous equations. This
subject is spread out over the book. Most of what we'll do is
contained in §7.4 (“forced oscillations”) and
§8.2.
Reading:
- §8.2: Examples 1, 2, and 3.
- §7.4: Examples 1 and 2 are review from last
week's topic (Second order systems). Read these examples
(example 2 is a bit long, but you should at least be able to
find the eigenvalues and vectors of the 3x3 matrix A by
yourself, and you should be able to interpret the corresponding
terms in the general solution). Then read page 440/441. I
will do an example like example 3 (p.441) in lecture this week.
Problems:
- §7.4: 8, 10, 12.
- §8.2: 1, 2, 5, 7, 8, 10.
- §7.4: This extra
problem about §7.4.
Due Monday April 25.
Reading:
- §7.1: Examples 3 and 4 show you how to turn one or
more higher order diffeqs into a system of first order equations.
Example 5 shows how to do the opposite.
- §7.1: The two paragraphs at the bottom of page 400
explain (very briefly) what a direction field and what
the phase plane portrait of a system of two first order
differential equations is.
- §7.4 The paragraphs on page 434, beginning
with Solution of Second Order Systems explain how to solve such
systems. In the special case that the matrix only has negative
eigenvalues Theorem 1 summarizes the result. It is worth knowing the
derivation since it also applies to variations on this kind of problem
(e.g. if you add friction or a magnetic field), while Theorem 1
doesn't tell you anything about slightly different equations.
I will go over this in class on Wednesday.
Problems:
- §7.3: 38, 39.
- §7.3: Read example 4, and in particular the explaining
paragraphs before that example, on page 427. Find the eigenvalues
corresponding to the system (22) (page 427) in which
k1=k2=1, and k3=K (K is some positive
number). For which values of the constant K do you get complex
eigenvalues, and for which do you get only real eigenvalues?
- §7.4: 1, 3, 5, 7.
Due Monday April 18.
Reading:
- §7.3: Read the entire section carefully. It explains how
to use the eigenvalues and eigenvectors to compute the general solution
to almost any system of linear ordinary diffeqs with constant
coefficients.
Problems:
- Write a description of the procedure which allows you
to compute the general solution to a system of
differential equations x'=Ax, where A is a real
n×n matrix. Your writeup should be clear enough
so that someone who knows how to compute eigenvalues
& eigenvectors of a square matrix could read it and
solve x'=Ax by following your description.
- §7.3:
3, 6, 9, 11, 19, 22, 24, 25.
This is not a very long list of problems,
but they are all multistep problems so don't underestimate the amount
of work needed.
Due Monday April 11.
Reading:
- §6.1: Read the entire section carefully. It explains
what eigenvalues and eigenvectors are, and how to compute
them. The Algorithm on page 369 summarizes how you go about
finding eigenvalues/vectors; the examples in this section show you
the details of the process.
- §6.2: Summary: An n-by-n matrix is
diagonalizable if you can find n linearly independent
eigenvectors for the matrix. So the question “can the
matrix A be diagonalized?” is the same as “can
you find n linearly independent eigenvectors
for A?” Eigenvectors corresponding to different
eigenvalues are always linearly independent.
Problems:
- §6.1: 6, 10, 14, 18, 19, 23, 29, 32, 33, 34.
- §6.2: 1, 5, 8, 12, 13, 14, 21, 22.
Due Monday March 28.
Reading: Much of the material here is review from math 222.
New are the concept of linear independence of functions (which I
won't emphasize) and the Wronskian.
- §5.2: Linear independence is the concept which
tells you when n solutions y1, …,
yn (of a linear homogeneous equation) are good enough
to generate the general solution via y =
c1y1 + … +
cnyn. The formal definition is given in
the book on page 305. There, and on the following page (306) it
is related to the Wronskian. See examples on pages 306, 307 for
the use of the Wronskian
- §5.2:
Page 310 explains how to solve nonhomogenous equations.
- §5.3: If you have forgotten how to solve
constant coefficient linear equations (in particular, if you
need to be reminded on what to do when there are complex roots)
all you need to know is in this section. Examples 4, 5, 7, and 8
(pages 321–323) are typical of what you need to know.
- §5.6: Resonance is an important example of how
solutions of a linear differential equation depend on the
frequency in the nonhomogeneous term. Think of this chapter
as a very long worked example of the theory explained in math
222, and in the previous sections in the book.
Problems:
- §5.2: 13, 15, 19. In addition to doing these
problems, compute the Wronskian for the given solutions in each
case. Finally, which of these equations has constant
coefficients?
- §5.2: 21, 24, 31. Also 25: write out a neat proof in
english (sentences that have a verb) beginning with “we must show
that ”, followed by “Proof:” The
coefficients p and q in this problem are functions
of x, so p=p(x), q=q(x).
- §5.3: 9, 15, 17. Also compute the Wronskians
of the solutions you find.
- §5.3: 27, 33, 36.
- §5.6: 4, 11, 12 (try to find particular
solutions of the form A cosωt + B sinωt,
as explained in example 6, page 360/361.)
Due Monday March 21.
Reading:
- §2.2: Make sure you understand the relations
between Figures 2.2.1 and 2.2.2 on page 93, and between figures
2.2.3 and 2.2.5 on page 95.
- §2.4: Make sure you understand Euler's method
(middle of page 113, until box on page 114.)
Problems:
- Explain in two or three sentences why the authors of the
book call equation (9) (page 95) the “explosion/extinction
equation.”
- §2.2: Find the equilibrium points, the
phase diagram, and use your phase diagram to determine
which of the critical points are stable, for the
differential equations in problems 1,2,3,4,7,9,10. You
do not have to find the general solution to these
equations (even though the book asks you to do this).
In each of these problems, let x(t) be the solution for
which x(0) = 0.73. Compute the limit of x(t)
as t goes to +∞. You can do this using the phase
diagram you found, and you do not need to compute the general
solution of the diffeq.
- §2.2: Draw Bifurcation
diagrams — more precisely, draw a diagram
as in example 6, but also indicate which of the
solutions are stable and which are unstable by
coloring the stable equilibria blue and the unstable
ones red (or your choice of colors): Problems 19,
20, 21, 29.
- §2.4: Do problem 4. You will probably need a
calculator. Do all computations in at least three decimal
places.
- Go to the
the logistic equation solver at the Virtual Math Museum.
The equation being solved is y' = ay(b-y) with a=1 and
b=5. Select "Euler's method" and "Connect dots", and set the
step size to 2.0 (instead of the default value 0.1). Plot a few
solutions, choosing initial values somewhere between 0 and 5. What
is most noticably wrong with the solutions produced by Euler's
method at step size h=2.0? (Do the solutions converge as
time goes to ∞?)
Due Monday March 7.
Reading: §4.6. and Chapter 1 (much of which should be review from calculus).
Problems:
§1.4: 19,20,21.
§1.5: 3, 5, 16[find two different solutions — the
equation is both linear and separable], 20.
§1.3: (do this problem after wednesday's lecture) In problem 3
you are shown a slope field for a diffeq.
- Two solutions are drawn in the picture: let's call them y1 and
y2. Are they equal at x=-3, ie. is y1(-3) =
y2(-3)? (you should answer this question without
solving the equation. If you use a theorem, say which one it
is, and quote the page in the book where you can find the
theorem.)
- Find the general solutions of the diffeq.
- Which values of the constant in your general solution
correspond to y1 and y2? How much do
y1(-3) and y2(-3) really differ?
(From the picture you can see that y1(3)≈1.7
and y2(3)≈-2.2.)
Due Monday February 28.
Reading: §4.3, §4.4, §4.5.
Problems:
§4.2: 2, 3, 5, 6, 11, 12, 17, 21.
23, 26, 27, 28.
§4.3: 9, 10, 17, 18, 19, 20
§4.4: 1, 2, 3, 4, 9, 12, 15, 22.
“Due” Monday February 21. This week
you get a reading assignment: read all of
§4.2! The important concepts are the definitions
of Rn (page 239), of Subspace (page
241), also Theorem 1 (page 241), and examples 2, 3, 4
(page 242). In lecture last Wednesday I showed you what
some subspaces of
R3.
Due Monday February 14.
§3.6, problems 1—20 (many of these are easy!), 21 (but
never do 22—40); conceptual problems: 50—54.
Due Monday February 7.
§3.4, problems 1, 5, 7, 11, 12; 27, 31, 32, 33, (optional: 36, 37);
§3.5, problems 2, 8, 9, 11, 21, 23, 32, 34.
Due Monday January 31.
§3.2, problems 2, 4, 6, 11, 23, 24, 25, 26;
§3.3, problems 1, 3, 7, 17, 33, 34.
Due Monday, January 24.
§3.1 — 8, 11, 16, 19.
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