Schedule, Suggested Readings, homework assignments/solutions,
(occasional) lecture notes, etc
Winter 2020
Preliminary Course Outline:
- Introduction
- Types of differential equations
- Initial value problems versus boundary value problems
- What it means to be a solution, interval of existence
- Added twists caused by nonlinearities
- First-order ODEs: qualitative methods
- First-order separable ODEs: how to solve them
- Planar Linear Systems
- Second-Order Differential Equations
- Planar Systems
- Preliminaries from Algebra
- Planar Linear Systems
- Eigenvalues and Eigenvectors
- Solving Linear Systems
- The Linearity Principle
- Phase Portraits for Planar Systems
- Real Distinct Eigenvalues
- Complex Eigenvalues
- Repeated Eigenvalues
- Changing Coordinates
- Classification of Planar Systems
- The Trace
- Dynamical Classification
- Exploration: A 3D Parameter Space
- Higher-Dimensional Linear Algebra
- Preliminaries from Linear Algebra
- Eigenvalues and Eigenvectors
- Complex Eigenvalues
- Bases and Subspaces
- Repeated Eigenvalues
- Genericity
- Higher-Dimensional Linear Systems
- Distinct Eigenvalues
- Harmonic Oscillators
- Repeated Eigenvalues
- The Exponential of a Matrix
- Nonautonomous Linear Systems
- Nonlinear Systems
- Dynamical Systems
- The Existence and Uniqueness Theorem
- Maximal interval of existence
- Continuous Dependence of Solutions
- The Variational Equation
- Equilibria in Nonlinear Systems
- Some Illustrative Examples
- Nonlinear Sinks and Sources
- Saddles
- Stability
- Bifurcations
-
Global Nonlinear Techniques (as time allows)
- Nullclines
- Stability of Equilibria
- Gradient Systems
- Hamiltonian Systems
- The Pendulum with Constant Forcing
Readings:
Ideally, you will have had a chance to do the readings (at least at a skim level) before class.
After class, you would do the readings at a deeper level. Try all of the exercises. I expect that
you're at a level of sophistication where you can recognize when some subset of problems are
quite similar to one another. If you're able to do this mental "clumping" then work the problems
in order until you feel that you've mastered whatever it is that you think the author is trying
to get you to master. If you haven't exhausted the problems, now try the last problem in the clump.
If you can do it, then you can check that clump off your list.
Errors in
"Differential Equations, Dynamical Systems, and an Introduction to Chaos"
by Hirsch, Smale, and Devaney.
Jan 7 & 9 : What is a differential
equation? Types of differential equations.
Initial value problems versus boundary value problems.
What it means to be a solution, interval of existence.
Added twists caused by nonlinearities.
First-order ODEs: qualitative methods.
First-order separable ODEs: how to solve them
Please read sections 1.1 and 1.2 in the book as well
as the "Definitions" section and "Final Thoughts" section in
of
Chapter 1 of
Paul's Online Notes on ODEs.
First homework set
Due Jan 25. Here are the
solutions, including four
different solutions to the eighth question.
First Webwork Assignment (due Jan 15) both
without and
with
answers. Each
student got a different assignment but they were all roughly
similar.
Jan 14 & 16 : Chapter 2 --- Planar linear systems.
Chapter 3 --- Phase Portraits for
Planar Systems:
distinct real eigenvalues.
For additional material, see
Chapter 5 of
Paul's Online Notes on ODEs.
Jan 21 & 23 : Chapter 3 --- Phase Portraits for
Planar Systems: complex eigenvalues, repeated eigenvalues.
Chapter 4 --- Classification of Planar
Systems.
Second Webwork Assignment (due Jan 22) both
without and
with
answers. Each
student got a different assignment but they were all roughly
similar.
Second homework set
Due Feb 8.
Here are
solutions and sketches of solutions. Please bear in mind
that some of the given information is more of a sketch/scaffold upon which
a complete solution could be built. (Specifically, would not be given full
credit if provided on an exam.)
Jan 28 & 30 :
Chapter 4 --- Classification of Planar Systems.
Third Webwork Assignment (due Jan 29)
without and
with
answers. Each
student got a different assignment but they were all roughly
similar.
February 3 First midterm exam. Held in BA1130. You can choose
between starting at 5:10pm and finishing by 7 or starting at 6:10pm
and finishing by 8. The midterm will cover chapters 1 (excluding
section 1.5), 2, 3 and section 4.1 of the text. The material from
webwork and the HW will also be fair game. Material that I discussed
in class will also be fair game. Here's the current version of the
help page that will come with the midterm.
Here are
the solutions to the midterm.
Fourth Webwork Assignment (due Feb 8)
without and
with
answers. Each
student got a different assignment but they were all roughly
similar.
Feb 4 & 6: Finish section 4.2. Sections 5.1 and 5.2 you should
read on your own. Section 5.3 Section 5.4 you should read on your
own. Section 5.5. Section 5.6.
Here's a
nice parametric curve plotter by Christopher Chudzicki
that can help you visualize solutions
in R^3. There are also demos for
parametric curves in R^2
and
parametric surfaces in R^3.
Fifth Webwork Assignment (due Feb 12)
without
answers. Each
student got a different assignment but they were all roughly
similar.
Here's the
worked through example of finding the matrix P that transforms
A into Jordan Canonical Form for a specific 4x4 matrix.
Feb 11 & 13:
Section 5.6, start chapter 6
Sixth Webwork Assignment (due Feb 26)
without
answers. Each
student got a different assignment but they were all roughly
similar.
Third homework set:
due March 1. Here are
solutions and sketches of solutions. Please bear in mind
that some of the given information is more of a sketch/scaffold upon which
a complete solution could be built. (Specifically, would not be given full
credit if provided on an exam.)
Feb 25 & 27:
Sections 6.4 and 6.5
Here's a
fuller explanation
of the proof of the lemma on pages 126 and 127
in the book. While the proof's technical, it's a very useful skill to learn
how to approach double series and how to "change coordinates" for series.
(I.e. how to view a double series using the same mentality and tools that
one would bring to a double integral.)
Fourth homework set:
due March 14. Here are
solutions and sketches of solutions.
Mar 3 & 5:
Sections 7.1, 7.2, 17.1, and 17.2
March 9 Second midterm exam. Held in EX300. You can choose
between starting at 5:10pm and finishing by 7 or starting at 6:10pm
and finishing by 8. New material being tested on: Section 4.2,
Chapters 5 and 6, and section 7.1. (But don't forget the old material,
of course!)
The material from
webwork and HWs 1, 2, and 3 will also be fair game. Problem 1 from HW4
is fair game. Material that I discussed
in class is fair game.
Here's the current version of the
help page that will come with the midterm.
Here are
the solutions to the midterm.
Mar 10 & 12:
Sections 7.1, 7.2, 17.1, and 17.2
Mar 17 & 19:
Sections 7.2, 7.3, 7.4, 17.3, and 17.4.
Class is meeting on zoom.
Fifth homework set:
due March 29. Here are
solutions.
Mar 24 & 26:
Sections 8.1, 8.2, and 8.3. Class is meeting on zoom.
Mar 31 & April 2:
Sections 9.1 and 9.2. Class is meeting on zoom.
Sixth homework set:
due April 20 (if you're handing it in).
Here are
solutions.
Information about the essay:
with deadlines of April 10, 17, 22, and 25
(if you're handing it in).
Seventh homework set:
due April 25 (if you're handing it in).
Here are
solutions.