Math 302:  Ordinary Differential Equations

Fall 2015 Course Page

Instructor:  Dr. Richard Brown

Sections: 

TA-run section meetings

Section 1:  Xing Wang, Tuesdays at 1:30pm, Latrobe 107

Section 2:  Xing Wang, Tuesdays at 3:00pm, Krieger 300

Section 3:  Harry Lang, Thursdays at 3:00pm, Krieger 308

Section 4:  Tianyi Ren, Tuesdays at 4:30pm, Krieger 300

Section 5:  Chenyang Su, Thursdays at 1:30pm, Shriver 104

Section 6:  Chenyang Su, Thursdays at 3:00pm, Krieger 300

Section 7:  Harry Lang, Thursdays at 4:30pm, Dunning 206

Section 8:  Tianyi Ren, Tuesdays at 3:00pm, Mattin 162

Text:

Elementary Differential Equations, 10th Edition Boyce, William E. and DiPrima, Richard C. New Jersey: Wiley, October 27, 2008 ISBN-10: 0470458321 | ISBN-13: 978-0470458327

 

Course Syllabus and Homework Assignment Schedule

 

There are Java Applets that are quite useful in understanding the nature of Ordinary Differential Equations.  They can be found here: 

 

JAVA Applets for ODEs

 

For now, pay attention to the Slope Field Calculators, particularly this framed version, which will load in a separate window.  I suggest you play with these applets.

 

Documents of interest:

• EXAM Schedule for Friday, October 12, 2012:  Go to the room corresponding to your section number, not necessarily your TA.

  • Sections 1 and 2:  Krieger 205, noon – 12:50pm (Wang, Su)

  • Section 3: Barton 117, noon – 12:50pm (Lang)

  • Section 7: Latrobe 107, noon – 12:50pm (Ren)

  • Section 5 and 6:  Krieger 205, 1:30 – 2:20pm (Wang, Su)

  • Section 4: Barton 117, 1:30 – 2:20pm (Lang)

  • Section 8: Latrobe 107, 1:30 – 2:20pm (Ren)

• Exam A Solutions, Exam B Solutions, Exam C Solutions, Exam D Solutions

• Worksheet on Second Order ODEs:  Resonance

• Notes from class:

  • October 15, 2012:  Sections 4.1 and 4.2

  • October 17, 2012:  Beginning of Section 7.1

  • October 19, 2012:  Sections 7.1 and the beginning of 7.3

  • October 22, 2012:  Sections 7.3 and the beginning of 7.4

  • October 24, 2012:  Sections 7.4 and the beginning of 7.5

  • October 26, 2012:  Section 7.5

  • October 29, 2012:  No class!  Blame Sandy....

  • October 31, 2012:  Section 7.5

  • November 2, 2012:  Section 7.5 and 7.6

  • Lost Lecture:  Sandy's Notes, Section 7.7 notes on the Fundamental Matrix

  • November 5, 2012:  Sections 7.6 and 7.8

  • November 7, 2012:  Sections 7.8 and part of 9.1 with a review of the classification of linear equilibria.

  • November 14, 2012: Section 9.3.

  • November 16, 2012:  Section 9.3 and 9.7.

  • November 19, 2012:  Sections 9.7 and 8.1.

  • November 26, 2012:  Sections 8.1 and 6.1.

  • November 28, 2012:  Sections 6.1 and 6.2.

  • November 30,2012:  Section 6.3 and 6.4.

• Resonance Solution

• The Pendulum's equilibria....

• My Teaching Philosophy -- This is not really a part of the course.  I thought I would give you a sense for what my motivations are for teaching.  This will give you a sense for who I am. 

• Example Problems:

  • Section 2.1: A Linear ODE.  Another Linear ODE.

  • Section 2.2: A separable 1st-order ODE.

  • Section 2.6: Exact ODE example.  Not all exact, first-order ODEs are separable, but ALL separable ODEs are exact. 

  • Section 2.6:  Yet another Exact ODE.  This one from class....

  • Section 3.2:  Some words on the Wronskian determinant.

  • Section 3.2:  A Wronskian theorem;  It's either always 0 or never 0.

  • Section 7.5:  Some Mathematica on phase space.

  • Section 9.3:  Some solution curves (as level sets of a function in Mathematica) of an almost linear system.

• Worksheet on the Existence and Uniqueness Theorem (not assigned, though).

• Challenge Problem Sets:  These sets are nothing but extra problems that I find interesting and useful.  They are not to be considered homework or material I expect you to be able to do.  They are just extra stuff for those who would like to see extra stuff.

  • Challenge set 1:  Week 5: Sections 2.6, 2.8 and 3.1.

  • Challenge set 2:  Week 6: Sections 3.2, 3.3 and 3.4.

  • Challenge set 3:  Week 7: Sections 3.5 and 3.6.

  • Challenge set 4:  Week 8: Sections 4.1 and 4.2.

  • Challenge set 5:  Week 10:  Sections 7.4, 7.5, and 7.6.

Notes after the lecture:

• September 10, 2012:  The idea of solving an ODE using an integrating factor may look weird, given the end of the lecture last.  But really, it is not.  Here is an idea for study:  take a look at the problems 1-12 in the back of Section 2.1. Don't worry about solving them. Simply take each one, put it into the standard form, identify p(t), and calculate the integrating factor. That simple exercise alone will help enormously with these kind of ODEs.

• September 12, 2012:  When separating variables in a first-order, separable ODE, the term on the Left-Hand Side (LHS) looks complicated, and there is a temptation to simply "cancel" the dt terms.  One cannot, although the effect would wind up correct.  See this writeup!

• September 12, 2012:  The book's treatment of the structure and analysis of separable ODEs is different form the lecture (my take).  Seek to understand both, and, if successful, you will see these kinds of problems to be rather simple.

• September 17, 2012:  The criteria for establishing the existence and uniqueness of solutions to a first order Initial Value Problem are something to remember:  The main points to consider here, though are the following:

  • Even if solutions are not guaranteed to exist or be unique, they may be.

  • When solutions are unique, the integral curves cannot cross or meet.  This will be important when studying how solutions behave in the future. 

  • The form for a solution to dy/dt = f(t,y), given by the Fundamental Theorem of Calculus, will be very useful in the future.

  Also, pay attention to the Logistic Equation in Section 2.5.  It will appear again.

• September 19, 2012:  Speaking of bifurcation diagrams (as in the last class), arguably the most interesting bifurcation diagram out there involves a discrete version of the logistic ODE of Section 2.5 and the last class: z' = rz(1-z). The discrete version is given by the function

   f(z) = rz(1-z), where r is a parameter and we iterate the map to construct the trajectories: z₀, z₁ = f(z₀), z₂ = f(z₁) = f(f(z₀)), and so on. The value of r changes the number and nature of the discrete version of an equilibrium solution. There are crazy bifurcation points for certain values of r, and places where the map is completely chaotic (chaos is a math term, and very precisely defined!). Here is a bifurcation diagram for r.... Enjoy!
• September 21, 2012:  I just placed your homework for this next week on the schedule.  There is also an extra problem posted.  The Fish Problem.  That will be due with the rest of the homework.  Enjoy!
• September 23, 2012:  I just posted a more detailed version of the Exact ODE example form class.  This one better details the explicit solution (and not simply the implicit one), along with the domain and a check to see that it works.  Check it out here....
• September 25, 2012:  And here is the worksheet based on today's lecture on the Existence and Uniqueness Theorem for first-order ordinary differential equations.
• September 28, 2012:  Since I haven't actually proved it yet, you can assume for now that that 2-parameter family of solutions we calculated for the second order linear ODEs with constant coefficients (with two non-real distinct roots of the characteristic equation) IS the general solution.  We will establish this next time.
• September 30, 2012:  The Wronskian determinant is a very useful tool for checking that two solutions to a second order linear homogeneous ODE constitute a fundamental set of solutions.  To see this, check out this problem.
• October 3, 2012:  So now we know how to write out a fundamental set of solutions for a second-order, linear, homogeneous, ODE with constant coefficients when the characteristic equation has either two real, distinct roots, or complex (conjugate) roots.  The third case..?
• October 4, 2012:  I put up the Wronskian Theorem proof that I like.  Give it a read.  It's not long.
• October 5, 2012:  So the exam will cover all sections covered from the beginning of the course until Section 3.4 Repeated Roots:  Reduction of Order.
• October 8, 2012:  Now for the case where the ODE is non-homogeneous? 
• October 17, 2012:  Here, we finished up the discussion of nth order linear ODEs, showing just how straightforward they are to solve given the second order case.  Then we branched into systems of first-order ODEs; The notable thing here being that they generally show up in applications where there is more than one dependent variable to measure and study, and the equations describing the behavior of the derivative functions usually involve instances of all of the dependent variables (coupling).  Also, since ANY nth-order ODE can be systematically written as a system of n first order ODEs, a thorough study of systems of ODEs or first order would be sufficient to understand ODEs in general, no?
• October 19, 2012:  Starting with a general discussion of existence and uniqueness of systems of first-order ODEs, we segued into what linear systems look like in general (the non-homogeneous kind), and then into a bit of basic linear algebra.  We will need this for the more detailed discussion we will have on linear, homogeneous systems with constant coefficients, the basic framework for almost all non-linear systems theory we know!
• October 22, 2012:  Back to ODEs today.  Keep the notion that the data one obtains from a matrix (its determinant, eigenvalues and eigenvectors, for example) will be useful for solving systems of linear differential equations, at least when the coefficients are constants.  
• October 24, 2012:  Pay attention to the calculations of the eigenvalues and eigenvectors, and practice these.  At least for the 2X2 case, these calculations are not difficult, and will say ALOT about solutions to the ODE systems.
• October 31, 2012:  Start building your library of phase portraits for various linear 2x2, homogeneous, constant coefficient systems.  They will be our base examples for some non-linear analysis later.  Today, we will almost finish the distinct eigenvalue, or repeated eigenvalue with enough eigenvectors cases.
• November 7, 2012:  The review of the phase plane cases I did was quick, but provides the basic classification of equilibrium solutions at the origin for linear two dimensional systems.  Keep these in mind as we move into Chapter 9.
• November 14, 2012:  Classifying non-linear system equilibria is the focus for today's lecture.  Here is a small writeup about the Pendulum.  More examples to come.  And here is a plot of the level sets of the function of both x and y that solves the system we did in class.
• November 16,2012:  We complete Section 9.3 today.  Sections 9.4 and 9.5 are application section dealing with some very interesting biology models;   competing species and predator-prey models.  I will offer some homework on these but really, they are only examples of the material of Section 9.3.  We also will work through parts of Section 9.7 also, skipping 9.6. 
• November 19, 2012:  As much as we will do in Section 9.7, we will complete today and then move into some basic numerical methods for solving ODEs in Section 8.1.  The latter is the basic idea how computers create and graph solutions.  While you will never find yourself actually using these methods in real life, computer models that you will use will use them, and a basic understanding of their strengths and weaknesses is necessary to use them effectively. 
• November 26, 2012:  There is so much to say about the numerical solutions to ODEs.  Way too much for a course like this.  Here we will only focus on the most basic Euler Method, the subject of Section 8.1.  Scientists and engineers should read and think about the rest of Chapter 8 also.  It will be necessary if you will be using computers to solve ODEs that you know what they are doing and just how badly they are at actually solving them...
• December 7, 2012:  Here are the exam solutions for the two exams (Exam C and Exam D) I gave on Wednesday.  These are handwritten.  Sorry about that.  They should still be useful.  Please let me know what questions you may have.

This page will be updated regularly when new information about the course arises.  General information about course structure, requirements, as well as specific information related to your lecture or section, will be posted here and updated as needed. 
 

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