 |
Pendulum without damping: vector field and some
trajectories in the phase plane. Note the three critical points at
(0,0), ( ,0), (2,0). |
MATH 246H, Spring 2010: Differential Equations for Scientists and
Engineers
News
-
The FINAL EXAM will be on Thursday, May 13, 1:30-3:30pm.
Note the ROOM: MATH 0201 (NOT THE USUAL ROOM!!!)
- Critical Points of Autonomous Systems: Type and Stability
- Assignment 2 is due Tue, May 11. You can either hand the assignment
in Monday in class, or you can slide it on Tuesday under the door of my office (MTH 4409).
Use the instructions in Using Matlab for Autonomous Systems.
Problem 1: It should say y' = Ay.
It should say: initial points (y10,y20)
with
-3≤y10≤3, -3≤y20≤3.
See the sample code
where initial points (y10,y20) with
-7≤y10≤5, -7≤y20≤1 are chosen.
For problems 1(b), 3(a) use the code for plotting the direction field
and several trajectories together.
For problem 2 use the code for investigation of critical points.
Problem 3: Use the system of ODEs from problem 2.
Problem 3(a): Use initial points (y10,y20)
with
0≤y10≤3, 0≤y20≤2.
Proceed as in the sample code
where initial points (y10,y20) with
-7≤y10≤5, -7≤y20≤1 are chosen.
Problem 3(b): FIRST use the plot from 3(a) and DRAW BY HAND a curve
dividing initial points with different limits for t->infinity.
THEN make a second plot where you use ode45 to draw this curve, as explained in the problem.
- Please read Solving ODEs with the Laplace Transform in Matlab
and try out the commands on a computer
- Exam 4 was on Monday, April 26. It covered sections 3.6-3.8, 6.1-6.4.
average = 82, median = 85
25% of the class had score ≥ 93, 50% had score ≥ 85,
75% had score ≥ 71.
- Exam 3 was on Wednesday, April 7. It covered sections 8.6 and ode45, 3.1-3.5.
average = 87, median = 92
25% of the class had score ≥ 96, 50% had score ≥ 92,
75% had score ≥ 82.
- Read Using Matlab for Systems and higher order ODEs and try out
the commands on a computer. Note that you first have to download the file
vectfield.m and put it in
the same directory as your other m-files
Example from class:
f = inline('[(1-y(2))*y(1);(-1+y(1))*y(2)]','t','y')
[ts,ys] = ode45(f,[0,20],[2;1]);
figure(1); plot(ts,ys)
figure(2); plot(ys(:,1),ys(:,2)); hold on
vectfield(f,0:.2:2,0:.2:2); hold off
- Assignment 1, due Fri, March 12 in class
You can work in a team of up to three students
Additional Information needed for Assignment 1
- Matlab solutions for Exam 1, generated with
publish('e1hsol') from m-file e1hsol.m
- Read Using Matlab for First Order ODEs and try out
the commands on a computer. Note that you first have to download the file
dirfield.m and put it in
the same directory as your other m-files for the homework.
You should use these commands for solving differential equations and plotting direction fields and solutions
in the Matlab assignment.
- You should have done the Matlab practice problems by now.
Solutions are now provided!
- Exam 2 was on Monday, March 8. It covered 2.3, 2.6, 2.7, 8.1, 8.2, M5.3 .
average = 89, median = 91
25% of the class had score ≥ 96, 50% had score ≥ 91,
75% had score ≥ 86.
- Exam 1 was on Monday, February 22. It covered 2.1, 2.2, 2.5.
average = 84.9, median = 86
25% of the class had score ≥ 97, 50% had score ≥ 86,
75% had score ≥ 77.
- Exam 3 is moved to Wed, April 7.
- Textbooks: There are two required books:
- "Elementary Differential Equations", 9th ed. by Boyce, diPrima
- "Differential Equations with Matlab", 2nd ed. by Hunt, Lipsman, Osborn, Rosenberg
You have three ways to buy them (prices for new books at the campus book store)
- $116.30: Unbound pages for both books, ISBN 978-0-4-7054107-4
- $212.80: Bundle of two bound books, ISBN 978-0-4-7059077-5
- $235.60: Both bound books bought separately, ISBN 978-0-4-7003940-3 and ISBN 978-0-4-7171812-3
- Read chapters 2, 3, 4
in "Differential Equations with Matlab" now and
try the commands on a computer (note differences between Matlab versions).
There is Matlab
Tutoring for MATH 246 available if you have questions.
- Until Feb. 15 do the following
Practice Problem Set (will not be collected):
in Problem Set A in "Differential Equations with Matlab"(p. 47):
- problem 3 (do (a), (b) with both ezplot and plot),
- problems 5, 7, 8,
- problem 11 (in (b) use ezplot twice with hold on,
do not use the parametrization)
Solutions: (using "publish" in Matlab with m-files given below)
problem 3
(m-file),
problem 5
(m-file),
problem 7
(m-file),
problem 8
(m-file),
problem 11
(m-file).
- What I did in class: (numbers are sections in Boyce/diPrima, sections in "Diff. eq. with Matlab" are marked with "M")
- 01/25: 1.1
- 01/27: 1.2, 1.3, 2.1
- 01/29: 2.1
- 01/29: 2.1
- 02/01: 2.4 existence, uniqueness, interval of existence for linear ODE (p.68, 69) (you can do problems 1,2,3 on p. 75 now)
2.2 is done different than in the book:
ODE y'(t)=F(t)G(y) has 2 types of solutions:
(1) solutions y(t) with G(y(t))≠0 are found by separation
(2) in addition, there are constant solutions y(t)=c with c such that G(c)=0
both (1), (2) together form general solution of ODE
- 02/03: 2.4: interval of existence for nonlinear ODE
- 02/15: 2.4: existence and uniqueness theorem 2.4.2, example y'=y1/3
- 02/17: 2.5: autonomous ODE, equilibria, stability
- 02/19: review for exam, 2.3 (mixing problem)
- 02/22: exam 1
- 02/24: 2.6 (exact ODEs)
- 02/26: 2.7, 8.1: Euler method
- 03/01: 8.1 local truncation error, global error, 8.2: Improved Euler method
- 03/03: M5.3: Stability of Differential Equations
- 03/05: review for exam
- 03/08: exam 2
- 03/10: convergence proof for Euler method, amplification factors
- 03/12: stable and unstable problems, adaptive step size control
- 03/22: 8.6: Systems of ODE, Euler, Improved Euler, ode45, phase plane plot
- 03/24: 8.6: Euler, Improved Euler, ode45 for 2nd order ODE
3.1: homogeneous linear 2nd order ODE with constant coefficients
- 03/26: 3.4: repeated roots, 3.3: complex roots
- 03/29: 3.5: Method of undetermined coefficients I: g(t) = a eαt,
g(t) = P(t) eαt with polynomial P(t)
- 03/31: 3.5: Method of undetermined coefficients II: g(t) = P(t) eαt cos(βt)
or g(t) = P(t) eαt sin(βt) with polynomial P(t)
- 04/02: review linear algebra: linear independence, singular and nonsingular matrices, determinant,
solutions of a linear system
3.2: linear 2nd order ODE y'' + py' + qy = 0:
linear independence of solutions y1, y2, Wronskian
W(t) = y1y'2 - y'1y2
Theorem about Wronskian: Assume p(t), q(t) are continuous for t∈(α,β),
and that
y1(t), y2(t) are two solutions of y'' + py' + qy = 0,
let t0∈(α,β). Then the following statements are equivalent:
(i) the solutions y1(t), y2(t) are linearly independent
(ii) W(t0) ≠ 0
(iii) W(t) ≠ 0 for all t∈(α,β)
- 04/05: review for exam
- 04/07: exam 3
- 04/09: y'' + p(t)y' + q(t)y = g(t):
homog. problem: reduction of order (3.4, p.170-171)
inhom. problem: variation of parameters (3.6)
- 04/12: Mechanical vibrations (hom. ODE) (3.7)
- 04/14: Forced vibrations (inhom. ODE) (3.8)
- 04/16: Laplace transform (6.1, 6.2): transforms of 1, eat, t eat, y'(t), y''(t)
solution of IVP using Laplace transform
partial fraction decomposition
- 04/19: transforms of tneat, eatcos(bt), eatsin(bt);
partial fraction decomposition with multiple roots
- 04/21: partial fraction decomposition with complex roots
piecewise defined functions (6.3, 6.4)
- 04/28: y'=Ay: real eigenvalues (saddle pt) (7.1-7.5, 9.1)
- 04/30: y'=Ay: real eigenvalues, complex eigenvalues (node, spiral pt, center) (7.6, 9.1)
- 05/03: y'=Ay: complex and double eigenvalues (spiral pt, center, proper node, improper node)
(7.6, 7.8, 9.1)
- 05/05: nonlinear autonomous systems (9.1-9.3)
MATH 246 Resources
Information about time & place, instructor, textbooks, syllabus,
grading policy, Matlab assignments, dates of exams, final exam.
This was handed out in the
first class.
Read section 4.4 ``Presenting Your Results'' in "Differential
Equations with Matlab". You will only receive credit for your homework if you
follow the following rules:
- Printouts of interactive input at the Matlab prompt
will not be accepted! You must prepare
an m-file
problemX.m containing
the commands for each problem.
- All printed numerical values must be labeled,
all graphics
must be labeled
- Print out only those values which were asked for in the
problem!
- Describe your observations and answer all questions asked in the
problem.
The easiest way is to use the publish command
(available in Matlab 7 or higher).
Your m-file contains the main title (starting with %%),
then a "cell title" (starting with %%) followed by
comment lines
(starting with %), followed by Matlab commands, followed by additional
"cells" in the same format.
The publish command then generates an html file (which you can
print from a web browser) or a Microsoft Word file containing input, output,
and graphics.
(see also p. 37, 41 in "Differential
Equations with Matlab").
Example: From the m-file publish_example.m we generate the web page publish_example.html with the command publish publish_example
If you don't use publish: For each problem hand in
- a printout of the file
problemX.m containing the Matlab
commands, and comments answering the questions asked in the problem. Also
include comments for each plot command indicating which of the attached
graphics it produced.
- a printout of the Matlab
output (e.g., using the Matlab
diary command). Use the
command echo on to print out all Matlab commands together
with the output.
- a printout of all graphics. Clearly label all graphs and put a
comment in the file
problemX.m so that it is clear which
Matlab command produced which graphic.
- additional (possibly handwritten) pages containing
answers to questions asked in the problem
Using Matlab for Differential Equations
You are expected to read these documents carefully and try out the
explained commands and examples on the computer (you can copy and paste the
commands from the browser into the Matlab window).