Syllabus
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Textbooks
Boyce and DiPrima, Elementary Differential
Equations, 9th ed.
B.Hunt et. al., Differential Equations with
MATLAB, 3d ed.
Contact
Room: Math 3317. Phone: (301)405-5152. email: mvy@math.umd.edu
Office hours: TuTh 11:15-12:15 pm and by appointment.
TA
Chunting Lu (ctlu@math.umd.edu) Office hours: 4:15-5:15 M,W. Rm math 4400
Chae Clark (cclark18@umd.edu) Office hours: Tu 3-4, W 1-2 Rm math 2121
Tutoring
Look at: www.math.umd.edu. Go to undergraduate/resourses.
Tests
Test 1.
Material: BdP, Ch. 2: Sections 2.1-2.6, 2.8. Skip integrating factors
for exact equations.
Test 2.
Material: BdP, Ch. 3 and Section 4.2 from Chapter 4.
Test 3.
Material: BdP, Ch.7: Sections 7.1- 7.8;
Ch. 9: Section 9.1.
ATTENTION. Final exam. Saturday May 11. 1:30-3:30.
Chunting's groups ARC 0204. Chae's groups Phy 1412.
Sunday alternate May 12. 1:30-3:30. Math 0303.
Please arrive at 1:20. Bring photo ID.
Homeworks and Quizzes
Matlab. Bonus problem : Problem Set A. Problem 5 - 5 points. Due February 1.
Study Chapters 1-4 of DE with
Matlab. See sample solution in Section 4.4. You work individually on Problem Set A.
You'll work in groups of 3 on the next Matlab assignments. Matlab HW are accepted only in printed form.
Not accepted on the web.
Recommended HW problems. MATLAB Info.
Section 1.1 Recommended HW problems:1-20.
Section 1.3 Recommended HW problems:1-20.
Section 2.1 Recommended HW problems:1-20.
Section 2.2 Recommended HW problems:1-20.
Section 2.4 Recommended HW problems:1-12.
Section 2.6 Recommended HW problems:1-16.
Section 2.2 Recommended HW problems:31-38.Required for quiz 2 and exam 1.
Quiz 1. February 5. Sections 2.1, 2.2.
Quiz 2. February 12. Section 2.6 - Exact equations( skip integrating
factors for exact equations). End of section 2.2 - homogeneous equations.
PROBLEM SET B. Due Friday Feb. 15 Problems 3,13,19.
Answer all questions. Detailed explanations are recommended.
In problems 13 and 19 plot the graphs of f(y),
find equilibrium solutions and determine their type and stability
based on the graphs. Use that information when you
analyze the vector fields.
Make teams out of 3 people within your section. Work together and submit
one solution signed by all members of the team.
Study Chapters 5,7 of DE with
Matlab. It is recommended to look at sample solutions at the end of
the Matlab Book.
Section 2.4 Recommended HW problems:1-12.
Section 2.6 Recommended HW problems:1-16.
Miscellaneous problems at the end of Chapter 2.
Recommended problems : 1-29.
Section 2.5 Recommended HW problems:1-15.
Section 2.3 Recommended HW problems:1-23.
Sections 2.7,8.1 and 8.2. Recommended HW problems:1-4.
Stability of ODE. Recommended Matlab book Section 8.4.
PROBLEM SET C. March 1. Problems 1,10,14.
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.
Quiz 3. Numerical methods/Matlab. Feb 26. Material: DE with Matlab Ch.5,
and Ch. 7-8,
BDiP Sections 2.7,8.1,8.2. The following examples illustrate the concept of
stability of ODE.
stability1 and
stability2.
Section 3.1. Recommended HW problems:1-24.
Section 3.2. Recommended HW problems:1-27.
Section 3.3. Recommended HW problems:1-24.
Section 3.4. Recommended HW problems:1-14.
Section 3.5. Recommended HW problems:1-18 solve, problems 19-26
find correct form, do not evaluate coefficients.
Section 4.2. Recommended HW problems:11-14, 16.
Quiz 4. March 14. Material:
BDiP Sections 3.1-3.5,4.2. For this quiz skip Reduction of order
in Section 3.4.
Section 3.7. Recommended HW problems:1-20.
Section 3.8. Recommended HW problems:5-12.
Section 3.6. Recommended HW problems: 1-20.
Section 3.4. Reduction of order. Recommended HW problems: 23-28.
Exam 2. April 4.
Material: BdP Ch. 3 and Section 4.2 from Chapter 4.
SAMPLE PROBLEMS for exam 2.
Section 3.2: 16,17,34. Section 3.3 : 24. Section 3.4: 26 (add
question : find general solution). Section 3.5: 18,21. Section 3.6:
15 (add question : find general solution). Section 3.7: 9.
Section 3.8: 18 (plot sample graphs in two cases : when frequency
does not equal 1, and when frequency equals 1). Section 4.2: 16.
Problem set D. Due March 29. 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.
Section 6.1. Recommended HW problems:1-20.
Section 6.2. Recommended HW problems:1-16.
Section 6.3. Recommended HW problems:1-24.
Section 6.4. Recommended HW problems:1-11.
Problem Set E. April 12. Problems 10,12,13(a-c). Answer all questions.
Quiz 5. April 16. Laplace transform. Sections 6.1-6.4 of BDiP
and Chapter 13 of the Matlab book (skip delta function ).
Training problems for quiz 5. Section 6.2 problem 22. Section 6.3
problems 12,14,21. Section 6.4 problem 10.
Section 7.1. Recommended HW problems:1-6,17,18,22.
Section 7.2. Recommended HW problems:1,6,10,11,22-26.
Section 7.3. Recommended HW problems:16-22.
Section 7.5. Recommended HW problems: 1-6,15,16,24-27.
Section 7.6. Recommended HW problems: 1-6,11-20.
Section 7.7. Recommended HW problems: 1-8,11,12.
Section 7.8. Recommended HW problems: 1-4,7-10.
Section 9.1. Recommended HW problems: 1-16 .
Problem set F. Due May 3. 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.
Test 3. April 30.
Material: BdP, Ch.7: Sections 7.1- 7.8. Chapter 9 :Section 9.1.
Types and stability are reviewed in table 9.1.1.
Training problems for test 3.
As we did in class: Find general solution,
solve initial value problem, sketch the phase portrait, in the phase plane
sketch the solution of IVP, sketch graphs x1(t), x2(t).
Several problems in BdP when combined ask same questions (but add x1(t),x2(t)).
In particular the following problems cover all types (some of them we did
in class). Section 7.6 problems 9,3,1 correspond to Section 9.1 problems 5,6,7.
Section 7.5 problem 1 corresponds to Section 9.1 problem 1.
Add same questions as in Section 9.1 to problems 5 and 6 of section 7.5.
Combine section 7.8 problem 1 with Section 9.1 problem 9, and
add similar questions to problem 3 of section 7.8.
Consider several examples from above and for each of them
find matrix exponential exp(At).See section 7.7 and class exercises.
ATTENTION. Final exam. Saturday May 11. 1:30-3:30.
Chunting's groups ARC 0204. Chae's groups Phy 1412.
Sunday alternate May 12. 1:30-3:30. Math 0303.
Please arrive at 1:20. Bring photo ID.
Remaining Recommended HW.
Section 9.2 Problems 5-16. Do (a) and (c). In (c) determine type and
stability as we did in class..
Section 9.3 Problems 5-18. Table on page 513.
Section 9.4 Problems 1-6.
Section 9.5 Study example 1. Problems 1-5.
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