## Textbooks

C. David Levermore, Ordinary Differential Equations,see
https://courses.math.umd.edu/math246/NODE/1920F/main.html
login with your university of MD account
B.Hunt et. al., Differential Equations with
MATLAB, NEW EDITION COMING SOON.
## Contact

Room: Math 3317. Phone: (301)405-5152. email: mvy@math.umd.edu
Office hours: Tu,Th 11:15-12:15, Tu after lecture, and by appointment.
## TA

## Discussion Sections

On Friday Sections 0311 8-8:50, 0321 9-9:50, 0331 10-10:50, 0341 11-11:50
all in Math 0401.
## Tutoring

Look at: www.math.umd.edu. Go to undergraduate/resourses.
## Grading

There will be a total of 700 points available in the course,
allocated as follows.
Three exams, each worth 100 points, see dates in Canvas.
Several quizzes, each worth 25 points, with four best counted for
a total of 100 points, see dates in Canvas.
Matlab assignments, each worth 20 points with five best counted for
a total of 100 points.
Cumulative final exam worth 200 points.
## Missed exams, quizzes

Please note that for a missed exam a written explanation from a doctor
will need to be
provided. With the proper explanation 1/2 of your score for the Final Exam
will substitute the score for the missed exam.
For a missed quiz with the proper explanation 1/4 of your score for
the next exam will substitute the score for the missed quiz.
ATTENTION. FINAL EXAM. Wednesday December 11, 1:30-3:30.
Please arrive at 1:20. Bring photo ID.
ROOM ASSIGNMENTS.
Maps and 3 letter building codes can be found at
www.umd.edu/CampusMaps.
## Exams

Exam 1.
Material: Lvrmr, Ch. 1: Sections 1-6, 9.
See Training
Exam 2.
Material: Lvrmr, Ch. 2 Sections 1-5,6.1-6.4, 8.
Recommended to review for exam 2.
(a) Linear 1-st order equations (see Chapter I; (b) Wronskian (Section 2);
Constant coefficients (Section 6), typical problems 10-13,25 ;
Also look at internal link to Section 6 for summary of methods; (d)Applications (Section 8),
typical problems 2,11,20,22. Review types of motion: harmonic,
underdamped, overdamped, critically damped, resonance, steady state.
Exam 3.
Material: Lvrmr, Ch.3 Sections 1-5,7,9.
To prepare for exam 3 use problems from Section 4 Ex. 10-18,
Section 5 Ex.13-28, Section 7 Ex.9-26. In each problem you should not only
answer the questions of that problem, but answer all set of questions
similar to questions of Quizes 6,7 : eigenvalues, eigenvectors,
matrix exponential, solution of IVP, phase portrait, sketch solution of IVP
in the phase plane. Choose yourself initial conditions,
for example (1,0) or (0,1) or (1,1) or (1,-1) etc.
Section 9 Ex. 1-18.
Sections 8,10
are on the final but not on exam 3.
ATTENTION. FINAL EXAM. Wednesday December 11, 1:30-3:30.
Please arrive at 1:20. Bring photo ID.
ROOM ASSIGNMENTS. BRB 1101
Maps and 3 letters building codes can be found at
www.umd.edu/CampusMaps.
ATTENTION. Review for the Final Exam at the last two lectures
Common review , to be announced.
## Homeworks and Quizzes

Matlab. All problem sets are in the Matlab book.
Problem Set A. Problems 5,8,9,10.
Starting from the Fall semester of 2019 we use a new edition of the Matlab book, to appear soon. If you bought the 3-d edition, please keep receipt, to exchange the book when the new edition comes. There is no difference in problem set A
except it is recommended to use 'fplot' instead of 'ezplot'.
Study Chapters 1-4 of DE with
Matlab. See sample solution in Section 4.4.
You work in 3 people teams on all Matlab assignments.
Matlab HW are accepted only in printed form signed by all members of the team.
Not accepted on the web.
Quiz 1. Linear, Separable equations.Chapter 1,
Sections 2 and 3.
Recommended HW problems.
Section 2 Recommended HW problems:1-15.
Section 3 Recommended HW problems:1-20.
Section 4 Recommended HW problems:1-12.
Section 5 Recommended HW problems:1-23.
Section 6 Recommended HW problems:1-17.
Section 7 Recommended HW problems:1-10.
Section 8 Recommended HW problems:1-21.
PROBLEM SET B. Problems 3,12,18,20..
Study Section 5 "Graphical methods" of Lectures.
Study Chapters 5,7 of DE with Matlab.
It is recommended to look at sample solutions at the end of
the Matlab Book.
Answer all questions. Detailed explanations are recommended.
In problems 12 and 18 in addition to the book's questions do also
the following. Use "solve" and/or "fzero" commands to
find equilibrium (stationary) solutions. Then use "plot" command to
plot the graph of f(y)
and check that zeros of the graph coincide with equilibria you found.
Then plot (by hand) the phase-line portraits,
and determine type (stability) of equilibrium solutions
based on the phase-line portraits. Use that information when you
analyze the vector fields.
Quiz 2. Chapter 1. Study Section 5- Graphical methods, especially
phase-line portraits, and graphs of solutions in (t,y) coordinates.
Section 6- Applications.
PROBLEM SET C. Problems 1,10,14. Study Section 7 of
Chapter 1: Numerical methods. Also
study Chapters 3-8 of DE with
Matlab. See sample solution of one of the problems at the end of
the Matlab Book.
Also see an example
example. This example will work if you include
in the same directory the function M-file myeuler
myeuler. See Section 8.2.1 of Matlab book.
Recommended HW problems, Chapter 2.
Section 1. 7-10.
Section 2. 1-25.
Section 3. 1-5,9,10.
Section 4. 1-19.
Section 5. 13-19.
Section 6. 1-27.
Section 7. 1-14.
Section 8. 1-23.
Section 9. 1-20.
Quiz 3. Material:
a) Chapter 1: Numerical methods: Section 7 of the textbook and
Chapters 3-8 of the Matlab book.
Quiz 4. Material:
Chapter 2. Sections 1-5,6.1-6.4.
Quiz 5. Material:
Sections 7.1-7.3. Typical problems 2,4,13.
Problem set D. Problems 3,4,5.
COMMENTS to Problem set D.
In problem 3 plot the graphs of the linear
approximation and of the actual pendulum .
One can estimate the period T of
nonlinear oscillations based on the
graphs . Notice that the period equals twice the distance between two
consecutive moments x1 such that y1(x1) = 0 .
In order to find such moments you can plot graphs
using the option ``axis''.
For example
plot(x1,y1(:,1))
axis([1.56 \ 1.58 \ -0.001 \ 0.001])
Next in order to find a root more accurately you can use the ``zoom''
feature on the ``Figure'' window of Matlab.
Remark. We use plot(x,y(:,1)) not plot(x,y), because
when we use plot(x,y), matlab plots not only y(t), but also
an extra graph of velocity dy/dt, which is not needed here.
If you want ode45 to do more precise calculations
you can use ``Options'' described in Section 7.3 of the Matlab book.
In problem 4 you can try
to increase accuracy when the initial speed equals 2.
In that case when time is large Matlab produces wrong graphs.
First do your computations with the
defaut accuracy, then increase it consecutively. Explain
the difference between graphs and why eventually the graphs become wrong.
Solve problem 5 using the following values of damping coefficient :
b = .5, 1, 2 .
In this problem you can use Simulink or you can use function m-files.
For example can use the following function m-file for the linear model:
function ode = F(t,y,unused,b)
ode = [y(2); -b*y(2)-y(1)];
and call it for example Flinear.m
After that file is saved in the same directory as the main m-file
you can use in the main m-file :
for b = [.5 ,1 ,2 ]
[t,y]= ode45('Flinear.m', [0 20], [0 4],[ ],b);
plot(t;y(:,1))
end
Overall there are several possibilities for bonus in that Project.
Systems. Chapter 3. Recommended HW problems.
Section 1. Recommended HW problems:1-10.
Section 2. Recommended HW problems:1-13.
Section 3. Recommended HW problems:1-7, 14-17, 20-22.
Section 4. Recommended HW problems:2-15,18.
Section 5. Recommended HW problems:1-22.
Section 6. Recommended HW problems:1-23. In all problems determine type
and stability.
Section 7. Recommended HW problems:1-23.
Section 8. Recommended HW problems:1-18.
Section 9. Recommended HW problems:1-14.
Problem set F. Problems 1, 5.
When answering question 5(f)
classify the type and stability
of the critical points (0,0) and (pi,0).
COMMENTS to Problem set F.
Problem 1.
You can write general solutions by hand. When writing use
constants c1 and c2.
When answering question (b) for the first equation
answer an additional question. Let matlab solve initial value problem
with initial conditions x(0)=a, y(0) = b . Then matlab expresses solution
using constants a and b.
Note that you did the same using constants c1 and c2. Find relation between
constants (a,b) and constants (c1,c2).
Problem 5.
Question (c) is theoretical, just differentiate by hand.
You can express E either as a function of theta(t)( which is a
solution of the second order equation), or as a function of x(t),
y(t) , which are solutions of the respective system. When evaluating
dE/dt use the Chain Rule and after that use that theta (or x,y)
satisfy given differential equation.
Do the same when evaluating dE/dt in question (g).
In order to find b0 in question (f) do several approximations.
First you define inline function, corresponding to our system, call it
for example g. Then you can try something like
for b = 3:0.1:3.5
[t, xd] = ode45(g, [0 15], [0 b]);
plot (xd(:,1), xd(:,2))
and the graph shows where different trajectories go. You see
where
they diverge and based on that narrow the range of b. After you repeat
that procedure several times you can find the required b0 with good precision.