This is a (very) brief summary of what was (or will be) covered in each class.
December 8
Lecture Overview: For the nonlinear pendulum, discussed how the critical point at (0,0) changes to a spiral sink if we add friction; typically, the parameter c representing friction is taken to be positive, but if we take c < 0, (0,0) becomes a spiral source. Defined bifurcation and computed explicitly an example of a saddle-node bifurcation in which a system changes from having 3 critical points to having only 1. Introduced periodic orbits as a second type of invariant object (in addition to critical points) which can dominate dynamics in a nonlinear system. Did an example related to the Hopf bifurcation of an attracting limit cycle around a repelling critical point; defined alpha and omega limit sets and limit cycle. Proved Bendixson's criterion for the existence of a periodic orbit; this gives a necessary, but not a sufficient condition. Stated and briefly discussed the Poincare-Bendixson Theorem, which gives a sufficient condition for the existence of a periodic orbit.
The lecture material is partly contained in Sections 7.4 and 7.5 of your text. A good reference for Bendixson's Criterion and the Poincare-Bendixson Theorem is Nonlinear Differential Equations and Dynamical Systems by Verhulst.
Problem Set 4 will be accepted until Wednesday, December 15. See Problem Sets.
December 1
Lecture Overview: Classified critical points for 2x2 linear systems in the case of complex eigenvalues; defined stability and asymptotic stability; stability is related to whether or not the type of critical point in the linearized system persists in the nonlinear system: this depends on the real part of the eigenvalues; discussed why all types of critical points persist except the center, which may either persist or become a source or sink in the nonlinear system; did several examples of constructing phase portraits for nonlinear systems including a population model describing competition between two species and a predator-prey model; showed one approach to determining if a center persists using conservation of energy for the nonlinear pendulum.
The lecture material is contained in Sections 7.3 and 7.4 of your text.
Problem Set 4 is due on Wednesday, December 8. See Problem Sets.
November 17
Lecture Overview: Given a linear system with constant coefficients, showed how to find a sufficient number of linearly independent solutions when the characteristic polynomial has repeated roots and not enough eigenvectors; defined geometric and algebraic multiplicity of an eigenvalue; introduced generalized eigenvectors and eigenvector chains and showed how to use them to construct linearly independent solutions of the ODE. Introduced the study of first order, autonomous nonlinear systems via phase plane analysis; the idea is to find the critical points of the system and linearize around them to find the type of critical point; then we hope to pass information back to the nonlinear system. In the 2x2 case, discussed the three types of isolated critical points that can occur when both eigenvalues are real.
The lecture material is contained in Sections 11.4, 7.1, 7.2 and 7.3 of your text.
Problem Set 2 was returned and solutions to selected exercises were distributed.
Problem Set 3 is due today. Please try to get it to me before Thanksgiving break.
Problem Set 4 was distributed and is due on Wednesday, December 8. See Problem Sets.
November 10
Lecture Overview: There was a presentation of a shortcut theorem to partial fractions expansions given by Vinny Madera. We then continued with linear systems: Showed that we can solve a homogeneous first order linear system with constant coefficients using the eigenvalues and eigenvectors of the coefficient matrix; reviewed how to find eigenvalues and eigenvectors from Linear Algebra; discussed how to construct real solutions in the case of complex eigenvalues and eigenvectors. Introduced method of variation of parameters to solve non-homogeneous equations. Introduced matrix exponentials as a way to think about the solution to a linear system; this touches on matrix calculus, a very rich subject.
The lecture material is contained in Sections 11.1, 11.2 and 11.5 of your text.
Problem Set 3 is due on Wednesday, November 17. See Problem Sets.
November 3
Lecture Overview: Introduced the idea of an impulse function to model a strong force acting over a short period of time; we model this with a rectangular pulse of width epsilon and height 1/epsilon; as epsilon goes to zero, this rectangular pulse "converges" to the Dirac delta function concentrated at a single point; this is not a true function, but rather a distribution that acts on continuous functions. Proved a propostion showing how to calculate the integral of the delta distribution against a continuous function and used it to find the Laplace transform of the distribution; used this to solve a second order ODE with impulsive forcing. Introduced linear systems; showed that any higher order system or equation can be turned into a first order system and explained why that is an advantage; solved a simple linear system to introduce the idea of the phase plane or phase portrait of the system where we can draw the solution curves or trajectories; briefly introduced matrix-vector notation for the system.
The lecture material is contained in Sections 5.6 and 3.9 of your text.
Problem Set 3 is due on Wednesday, November 17. See Problem Sets.
October 27
Lecture Overview: Showed how the Laplace transofrm acts on higher order derivatives; used the Laplace transform to solve several examples of linear ODEs with constant coefficients; the main idea is that the transform converts a differential equation into an algebraic one that is linear in the unknown; we solve the algebraic equation and then must map back to a function in the original "t domain"; this last step of mapping back is often the most complicated as it involves taking inverse transforms of combinations of functions. Discussed discontinuous inputs, Heaviside and pulse functions, and their transforms; solved an application to an RLC circuit with discontinuous forcing. The Laplace transform gives us a unified way to solve such a problem which would otherwise have to be solved as a series of connected initial value problems.
The lecture material contains selections from Sections 5.3, 5.4, 5.5 and 5.7 of your text.
Problem Set 2 was due and will be accepted without penalty until Wednesday, November 3.
Problem Set 3 was distributed and is due on Wednesday, November 17. See Problem Sets.
Problem Set 1 was returned and solutions to selected problems were distributed.
October 20
Lecture Overview: Proved that Legendre polynomials are mutually orthogonal; defined the Gamma function and proved that it is defined for all x>0; computed the Gamma function for certain values, including integers and 1/2; derived Bessel functions of the first kind as solutions of a specific ODE; indicated other cases to consider that give rise to other types of Bessel functions. Defined the Laplace Transform; stated and proved a sufficient condition for the existence of the Laplace Transform for functions of exponential type; computed several examples. Proved several properties of the transform: Linearity, its relation to the derivative and integral; its action on convolutions of functions.
The lecture material is contained in Sections 4.4, 4.5, 4.6, and 5.2 of your text.
Problem Set 2 is now due on Wednesday, October 27. See Problem Sets.
October 13
Lecture Overview: Reviewed how to solve a linear ODE using power series centered at a point at which the coefficient functions are analytic; defined regular and irregular singular points; introduced the method of Frobenius which uses the roots of the indicial equation to generate series solutions centered at regular singular points; stated the Frobenius theorem; did examples which involved solving problems in two of the more complicated cases. Started looking at some special functions in the theory of ODE: Defined Legendre polynomials as solutions of a specific second order ODE.
The lecture material is contained in Sections 4.2, 4.3 and 4.4 of your text.
Problem Set 2 is now due on Wednesday, October 27. See Problem Sets.
October 6
Lecture Overview: Presented an extended application of a second order equation with constant coefficients modeling a mass on a spring; discussed damped versus undamped motion and free versus forced motion; discussed phenomenon of resonance and transient and periodic solutions; watched videos of the collapse of the Tacoma Narrows Bridge and the opening of the Millenium Bridge as examples of resonance gone awry. Began discussing power series solutions to linear ODEs; reviewed power series, radius and interval of convergence, Taylor series, analytic and singular points; please review these definitions and basic facts about series if they are not familiar to you; did an example of a solving a simple ODE using power series to introduce the basic ideas of the method.
The lecture material is contained in Sections 3.5, 3.8, 4.1 and 4.2 of your text.
Problem Set 2 is due on Wednesday, October 20. See Problem Sets.
September 29
Lecture Overview: Proved that if r is a root of multiplicity k of the characteristic equation associated with an nth order linear ODE with constant coefficients, then there are precisely k solutions of the form xjerx for j = 0,..., k-1; the fact that these are linearly independent is left for you to prove in Problem Set 2. Introduced two methods for finding solutions to the nonhomogeneous equation: undetermined coefficients and variation of parameters. The method of undetermined coefficients has the advantage of being relatively simple when the forcing function is of a particular form; variation of parameters works more generally, even in the nonconstant coefficient case, as long as we have the general solution of the associated homogeneous equation. Introduced Cauchy-Euler equation as an example of a higher-order nonconstant coefficient equation that we can solve.
The lecture material is contained in Sections 3.6 and 3.7 of your text.
Problem Set 1 was collected. I will accept it without penalty until Wednesday, October 6. See Problem Sets.
Problem Set 2 was assigned and is due on Wednesday, October 20. See Problem Sets.
September 22
Lecture Overview: Stated Existence and Uniqueness Theorem for nth order linear homogeneous equation; the proof is postponed until we introduce systems of equations; used this to prove the theorem involving the Wronskian from last week. Proved that an nth order linear equation has exactly n linearly independent solutions and that the general solution is the set of all linear combinations of these n solutions; proved that the solution set for a linear nonhomogeneous equation is simply the solution set to the associated linear homogeneous equation plus a particular solution; thus, the solution sets to linear differential equations follow exactly the same form as the solution sets from Linear Algebra. Introduced how to solve a linear equation with constant coefficients using the characteristic equation; proved that if the characteristic equation has n distinct roots, then the differential equation is completely solved by the set of associated exponential solutions; this essentially amounts to proving that the n exponential solutions are linearly independent via the Vandermonde determinant; discussed the case of complex roots and stated Euler's identity relating the complex exponential to sine and cosine. Began discussing the case of repeated roots and showed the method of reduction of order; this suggests that repeated roots lead to solutions of the form polynomial*exponenitial; we will prove this general fact next week.
The lecture material contains selections from Sections 3.3, 3.4 and 3.5 of your text.
Problem Set 1 is due on Wednesday, September 29. See Problem Sets.
September 15
Lecture Overview: Presented several more applications: (1) Using substitution to solve homogeneous equations; (2) derived the equation describing the arc of the cable of a suspension bridge; (3) discussed population models and solved the logistic equation. Discussed interpretations of these solutions in terms of the applications. Began discussing higher order linear differential equations; recalled some important facts from Linear Algebra, including the role of determinant in determining whether a linear system has a unique solution; described structure of solution sets to both homogeneous and nonhomogeneous equations; we will see the same structure emerging in our solutions to linear ODEs. Defined linear dependence and independence of sets of functions; introduced the Wronskian as a way to check that functions are linearly independent; in general, the converse does not hold, but for solutions of a linear ODE, the Wronksian does tell us when a set of functions is linearly dependent as well (proof postponed). Introduced operator notation for higher order linear equation; showed that the operator is linear; proved the Superposition Principle for solutions of homogeneous linear ODEs.
The lecture material contains selections from Sections 1.3, 2.3, 2.4, 3.1, 3.2, 3.3 of your text. The material on Linear Algebra is contained in Sections 10.4 and 10.5 of your text; see also Appendix B.
Problem Set 1 is due on Wednesday, September 29. See Problem Sets.
September 8
Lecture Overview: Introduced three methods of solution for first order equations: (1) integrating factors for linear equations; (2) separation of variables and the introduction of singular solutions; (3) exact equations, and integrating factors to make an equation exact. As an application, solved a problem involving Newton's Law of Gravitation and interpreted the solution; found escape velocity.
The lecture material is contained in Sections 2.2, 2.4 and 2.5 of your text.
The Introductory Questionnaire was collected during class. Please hand it in if you have not done so already!
Problem Set 1 was assigned and is due on Wednesday, September 29. See Problem Sets.
September 1
Reviewed course outline and policies on the course syllabus. This file and most others posted are pdf files.
Lecture Overview: Gave brief introduction to differential equations, defining the difference between ODEs and PDEs and the goals of modeling physical processes through differential equations: In this course we will focus on solving ODEs (and not deriving the equations from physical principles). Defined initial value problem for an nth order equation and solved several examples. Discussed local versus global existence and uniqueness of solutions via several examples: the example of water draining from a bucket following Toricelli's law is a good example to keep in mind. Proved existence and uniqueness of solutions of first-order equations (under suitable assumptions) using Picard's method of iteration.
The lecture material is partly contained in Chapter 1 of your text. A good reference for the proof of existence via Picard iteration is Chapter 1 of Theory of Ordinary Differential Equations, by Coddington and Levinson.
An Introductory Questionnaire was handed out during class which asks you about your knowledge of prerequisities for the course - specifically complex variables and linear algebra. Please bring the completed questionnaire with you to class on September 8.