MATH 246H, Spring 2013: Differential Equations for Scientists and
Engineers
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Phase portrait for Pendulum with damping
y'' + 0.2y' + sin(y) = 0
Note the two types of critical points:
asymptotically stable spiral points at (0,0), (2π,0), (4π,0) etc.
unstable saddle points at (π,0), (3π,0); (5π,0) etc.
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News
- Beginning Monday, June 3, 2013 the Math Undergraduate Office (UGO), Room 1117, will provide students with copies of their Spring 2013 final exams.
- All course results have been published. You should be able to see all your scores.
For the final exam the following cutoffs were decided by all MATH246 instructors:
0-94: F, 95-119:D, 120-154:C, 155-174:B, 176-200:A
- FINAL EXAM:
- Saturday, May 11, 1:30 to 3:30pm in MTH0106
- There will be a review session for all sections of MATH246 on
Friday, May 10, 10-12 in TYD 0130.
-
Note: I changed the recommended problems for 7.7, 9.1, 9.2.
do the recommended practice problems
for sections 1.1, 2.1, 2.2, 2.4, 2.5, 2.6, 2.3, problems for numerical methods,
sections 3.1, 3.4, 3.5, 3.6, 3.7, 3.8, 6.2, 6.3, 6.4, 7.5, 7.8, 7.6, 7.7, 9.1, 9.2, 9.3 !
- Assignment 3 was handed out on May 2, it is due on May 9
Problem (a): For (ii): first explain that y2(t)=0. Then you have for
y1(t) an ODE of the form y1' = G(y1). This is an autonomous
problem, so you can find the behavior of the solutions by looking at the graph of G.
For (i) you can proceed similarly.
Problem (c): For each starting point find the solution for t going from 0 to 100, and for
t going from 0 to -100. Use
warning('off','MATLAB:ode45:IntegrationTolNotMet')
at the beginning of your m-file to suppress all those warning messages.
Paint a part of the rectangle yellow by hand with a highlighter. For the starting points in the yellow region
both populations will die out as t goes to infinity. For the starting points in the remaining part of the
rectangle you will see a second type of behavior as t goes to infinity.
Problem (d):
The separatrix separates the yellow from the white region of the rectangle. The separatrix is a trajectory
which goes into a saddle point as t goes to infinity. Therefore we can plot it by starting close to the
saddle point, and letting t go to minus infinity.
Problem (e): Follow the instructions for investigating critical points.
- Using Matlab for autonomous systems: plot phase portrait, find critical
points, determine type of critical points
- How to find the type of critical points of an autonomous system. Note that in some
cases (equal eigenvalues, eigenvalues with real part zero) we cannot completely decide the type for the
nonlinear problem.
- Assignment 2 was handed out on April 25. It was due on May 2.
- Exam 3 was on April 23
- You can download the latest version of Matlab (R2013a) for free
from TERPware.
- Solving ODEs using the Laplace Transform in
Matlab
- Solution of Exam 2 with Matlab, generated with publish
from m-file exam2.m
- Exam 2 was be on Thursday, May 14. It covered
- Euler method and Improved Euler method for 1st and 2nd order ODE
- Using ode45 for 2nd order ODE
- linear 2nd order ODE with constant coefficients y'' + b·y' + c·y = g(t)
- case g(t) = 0
- case g(t) = a·eαx using method of undetermined coefficients
- Due date of Assignment #1 is extended to Thu March 7!
- Predator-prey example from class, generated with publish from the file
predprey.m.
You first have to download the file vectfield.m.
- Assignment #1 was handed out in class on Feb.21, due date extended to Thu, March 7 in class (not
March 4 as printed on the homework).
Instructions for Assignment 1
- Solution of exam using Matlab obtained with "publish" from e1.m.
- Try out the Matlab examples in the quick overview below
and read the detailed instructions. Note that you
first need to download the latest version of the file dirfield.m.
- Read chapters 2, 3, 4
in "Differential Equations with Matlab" now and
try the commands on a computer.
There is Matlab
Tutoring for MATH 246 available if you have questions.
- Until Feb. 12 do the following
Practice Problem Set (will not be collected):
in Problem Set A in "Differential Equations with Matlab"(p. 49):
- 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")
- 01/24: 1.1, 1.2, 1.3
explained "1st order ODE", "general solution", "initial value problem (IVP)", "solution of IVP", "direction field"
- 01/29: 2.1 (linear 1st order ODE)
- 01/31: Theorem 2.4.1 (p.68), 2.2 (separable equations), p. 72-75 ("interval of definition", "implicit solution")
Note for separable ODE y'=p(t)q(y):
- there are two types of solutions:
(i) constant solutions y(t)=c where c is a value with q(c)=0
(ii) solutions found by separation of variables: Integrate dy/q(y) = p(t)dt to get solution in implicit form, then solve for y.
- initial value problem: Use implicit form of solution to determine C, but if there are multiple solutions for y be
careful to pick the correct one.
- 02/05: 2.5, M7.3 (autonomous ODEs)
- 02/07: 2.6 (exact ODEs)
- 02/12: review of exact ODEs, 2.3 ("mixing problem")
- 02/14: applications (falling object p.2,13; Newton's law of cooling), existence and uniqueness of solutions of IVP (Theorem 2.4.2)
- 02/19: Exam 1
- 02/21: stable and unstable ODEs (M5.3), Euler method (8.1, M8.2.1)
- 02/26: error propagation for stable and unstable ODEs
- 02/28: Improved Euler and RK4 method, ode45 (8.2, 8.3, M8.2.2-8.2.4)
systems of first order equations and Euler, Improved Euler, ode45
(8.6)
- 03/05: 2nd order ODE: Euler method, impr. Euler method, ode45;
ODE y'' + b·y' + c·y = 0 with constants b,c (3.1, 3.4)
- 03/07: Complex Roots of char. equation (3.3)
- 03/12: Method of Undetermined Coefficients (3.5)
- 03/14: Exam 2
- 03/26: Method of Undetermined Coefficients (3.5)
- 03/28: Reduction of Order (p.170), Variation of Parameters (3.6)
- 04/02: Mechanical Vibrations: no damping, underdamping, critical damping,
overdamping (3.7)
- 04/04: Forced Vibrations: (i) no damping (resonance, beats), (ii) damping (3.8)
- 04/09: Laplace Transform (6.2)
- 04/11: Laplace Transform: Partial Fraction Decomposition for repeated and complex roots (6.2)
- 04/16: Laplace Transform: Piecewise defined forcing functions (6.4),
Impulse Functions (6.5)
- 04/18: Review for exam;
systems of ODEs y'=f(t,y);
autonomous system
y'=F(y): vector field,
critical points, stability of critical points (9.2);
linear ODE system with constant coefficients: y'=Ay (7.5)
- 04/23: Exam 3
- 04/25: linear ODE system with constant coefficients (7.2,7.3,7.5)
- 04/30: linear ODE system: repeated eigenvalues (7.8), phase portraits for real
eigenvalues (7.5, 7.8, 9.1)
How to find the type of the critical point
- 05/02: linear ODE system: complex eigenvalues (7.6)
summary: type of the critical point, how to sketch phase portrait
(9.1)
nonlinear autonomous system: find critical points, use Jacobian to
determine type of critical point at each critical point (9.3)
- 05/07: Review: 7.7, 9.1, 9.2; 9.3: more examples, 9.5
MATH 246 Resources
Syllabus
The syllabus contains information about time & place, instructor, textbooks, syllabus,
grading policy, Matlab assignments, dates of exams, final exam.
This was handed out in the
first class.
Note that Matlab was originally designed to do computations with numerical values (with about 16 digits accuracy),
as well as vectors and matrices of numerical values.
The Symbolic Toolbox (included in the student version) adds commands for operations with
symbolic expressions.
In this course we will need both numerical and symbolic commands. Some tasks
(such as plotting a function) can be performed either with numerical or with symbolic commands,
but in very different ways.
This is somewhat confusing at first.
- Using Matlab for autonomous systems
- Solving ODEs using the Laplace Transform in
Matlab
- How to solve systems and higher order differential equations with Matlab
- How to solve 1st order differential equations with Matlab:
- Quick overview
First download the file dirfield.m and put it in the same directory as your other m-files for the homework.
Example: Solve the initial value problem
y' = t·y2, y(-1) = 1
Plot direction field for
-2 ≤ t ≤ 2, -1 ≤ y ≤ 4
and plot solution for t from -1 to 2:
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f = @(t,y) t*y^2;
hold on; dirfield(f,-2:.2:2,-1:.2:4);
[t,y] = ode45(f,[-1,2],1); plot(t,y); hold off
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Find symbolic solution of IVP:
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dsolve('Dy=t*y^2,y(-1)=1','t')
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- More details: Using Matlab for 1st order ODEs
- Learning
Matlab:
- Where to use Matlab
- Common Problems with Matlab
- Matlab
documentation, Symbolic
Math Toolbox Documentation
