Ordinary Differential Equations
Schedule
Lectures. Wednesday 4-6 pm; Thursday 2-3 pm. To be delivered online.
Link will be sent through moodle or posted on this page!
Recitation instructor: Emanuel Sygal
Recitations.
Thursday 3-4 pm and 6-7 pm.
Emanuel's office hours. Sunday
12-1 pm.
Course grader: Ariel Rom
About this course
Topics include: Existence and uniqueness of ODEs. Exponentiating
matrices and linear ODEs. Bases of solutions and Wronskians. Method of
variation of parameters. Power series expansions. Qualitative theory of
non-linear systems. Boundary value problems. Sturm-Liouville theory.
Prerequisites
I expect familiarity with linear algebra and multi-variable calculus
(chain rule, equality of mixed partials).
References
There are many books on differential equations. The two books that I
found most helpful are:
Differential Equations, Dynamical Systems, and an Introduction to
Chaos by Morris W. Hirsch, Stepen Smale, Robert L. Devaney
Ordinary Differential Equations by Garrett Birkhoff and
Gian-Carlo Rota
At times, I will follow the beautiful notes by Shiri Artstein (in
Hebrew).
Homework
Assignment 1
Assignment 2
Assignment 3
Assignment 4
Assignment 5
Assignment 6
Assignment 7
Assignment 8
Assignment 9
Assignment 10
Assignment 11
Practice Slides
Practice Exam
Marking scheme
It is university policy that homework is to be submitted as if it were a regular semester.
You can submit it via Moodle to
Ariel Rom. I am told that there are many apps that allow you to make quality scans
on a modern smartphone. If you don't have a modern smartphone, you can latex your
solutions.
Week 1 in review
This week we discussed we studied several simple differential equations:
y' = y, y' = \sqrt{y}, y' = y^2, y' = y(L-y). We noticed that in the
first, third and fourth examples, the integral curves (solutions of the
equation) foliate the plane, that is, through every single point in the
plane, passes a unique integral curve. In particular, no two integral
curves intersect. This is a consequence of the Picard-Lindelof theorem
(the existence and uniqueness theorem for first-order ODEs) -- it is
enough that the function f in y' = f(x,y) is C^1. We will prove this
theorem in a week or two. We also saw that this property fails in the
second example.
We discussed the concept on an equilibrium
point (where y' = 0) and saw some examples of stable and unstable
equilibrium points.
We also discussed two second order systems:
the harmonic oscillator x'' = -x and x'' = -sin x. The idea was to
convert them into first-order systems by introducing a dummy variable y
= x'. In both examples, we defined an energy function E(x,y) and proved
that the energy is conserved under the dynamics, so that (x,y) travels
along the level sets of E.
Finally, we discussed the
inhomogeneous problem y' = y + f(x) and saw that it could be solved by
the method of variation of parameters: one examines the homogenous
equation y' = y, which has the solution Ce^x and made the constant a
function of x: C(x)e^x. Notes
Week 2 in review
This week, we have discussed the theorems of Peano and Picard-Lindelof concerning the
existence and uniqueness of ODEs. I expect that you understand the statements of the theorems
and have some idea how the proofs work, but if you don't follow the proofs 100%,
it is perfectly fine, since we will not need the ``details of the proofs'' in this course.
I apologize if the lectures were somewhat technical.
Notes 2 (updated)
Slides (Wed)
Slides (Th)
Week 3 in review
This week, we discussed the principle of continuation of solutions,
the concept of maximal/minimal solutions (when one does not have
uniqueness) and Osgood's uniqueness criterion (generalizing Picard-Lindelof).
We also discussed exponentiation of matrices.
Notes 3 (draft)
Week 4-5 plans
Notes (updated) and Slides
Week 6 plans
This week we will discuss the concept of distributions or generalized functions
and what it means for a differential equation to hold in the sense of distributions.
For the purposes of this course, the only non-trivial distribution is the Dirac delta
mass and one only needs to know that the presence of the delta mass indicates a jump
discontinuity. The notions of Greens functions and delta masses will not appear on the exam,
but may appear on the homework.
Notes (draft)
Further notes
Notes on higher-order equations
Laplace Transform and Slides
Notes on Power Series Solutions (not on the exam)
Fourier Series and the Heat Equation
Wave Equation, Sturm-Liouville Theorem
Exact Equations, Kepler's Laws