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Schedule
Notes
The schedule is tentative and may change as the course progresses.
For information about add/drop dates and other matters related to course registration, please consult the Registrar's Website.
All dates and times are in ET (Eastern Time).
Section numbers are as in Differential Equations and Boundary Value Problems (6th Edition) by C. Henry Edwards, David E. Penney, and David Calvis.
Please see the Homework tab for information about homework frequency. Below, O\(N\) refers to the \(N\)th online assignment (i.e., MyLab homework) and H\(N\) refers to the \(N\)th written assignment (i.e., Gradescope homework).
Week of August 25th (Week 1)
Monday, August 25th
Topics: Course Logistics; review of differential equations and linear algebra; systems of ordinary differential equations; review of matrix algebra; index formula for matrix multiplication; vector formulation of systems of ODEs; ansatz for solution; reducing to the eigenvalue-eigenvector problem.
Reading: §5.2 The Eigenvalue Method for Homogeneous Systems.
Video: (Kaltura01)
Wednesday, August 27th
Topics: General formulation of constant-coefficient, homogenous systems of ODEs; finding eigenvalues via characteristic polynomials; finding eigenvectors via row-reduction; finding solutions of a system from an eigenpair; the principle of superposition; general solution of a system with nondefective matrix; issues with the algorithm.
Reading: §5.2 The Eigenvalue Method for Homogeneous Systems.
Video: (Kaltura02)
Friday, August 29th
Topics: General solution to a system with a diagonalizable matrix; sketch of a \(3\times 3\) example; solving IVPs; systems with complex eigenvectors and eigenvalues; complex eigenpairs of real matrices come in complex conjugate pairs; Euler's formula; modified ansatz for complex eigenpairs; systems with defective matrices; review of defective matrices; algebraic multiplicity (\(k\)); geometric multiplicity (\(p\)); defect of an eigenvalue (\(d = k - p\)); analogy with \(n\)th order constant coefficient linear ODEs with repeated roots; rank-\(r\) generalized eigenvector; generalized eigenvectors come in sequences of decreasing rank with the same eigenvalue; modified ansatz for rank-\(2\) generalized eigenvectors.
Reading: §5.2 The Eigenvalue Method for Homogeneous Systems; §5.5 Multiple Eigenvalue Solutions.
Video: (Kaltura03)
Week of September 1st (Week 2)
Office Hours will be on W, 11:00 AM to 12:30 PM (in-person) and F, 11:00 AM to 1:15 PM (via Zoom) this week (in view of Labor day and a personal exigency).
O1 is due Wednesday, September 3rd at 11:00 PM.
O2 is due Friday, September 5th at 11:00 PM.
H1 is due Saturday, September 6th at 11:00 PM.
Monday, September 1st
Wednesday, September 3rd
Topics: Review of rank-\(r\) generalized eigenvector; an example of an eigenvalue with defect \(2\); brief description of the step-up algorithm for finding generalized eigenvectors; general ansatz for rank-\(r\) generalized eigenvectors; asymptotics of the function \(t \mapsto e^{\lambda t}\); phase planes for various systems \(2\times 2\) matrices.
Reading: §5.5 Multiple Eigenvalue Solutions; §5.3 Solution Curves of Linear Systems.
Video: (Kaltura04)
Friday, September 5th
Topics: Review of previous examples; table of behaviour according to eigenvalues; saddle point; (improper) nodal source; principle of time reversal; source versus sink; proper source and sink; example with many equilibrium solutions; spiral sink and source; determining directions by looking at the matrix action; center of ellipses.
Reading: §5.3 Solution Curves of Linear Systems.
Video: (Kaltura05)
Week of September 8th (Week 3)
O3 is due Wednesday, September 10th at 11:00 PM.
O4 is due Friday, September 12th at 11:00 PM.
H2 is due Saturday, September 13th at 11:00 PM.
Monday, September 8th
Topics: The case of defective matrices; nodes versus spiral points; autonomous systems; phase plane; direction field; nonlinear systems; critical points; stability of a critical point (stable and unstable).
Reading: §5.3 Solution Curves of Linear Systems; §6.1 Stability and the Phase Planes.
Video: (Kaltura06)
Wednesday, September 10th
Topics: stability of a critical point (asymptotically stable; stable but not asymptotically; unstable); long-term behaviour of trajectories; isolated critical points; Taylor's theorem; linearization; Jacobian; stability of linear systems; \((0,0)\) is isolated critical point of a linear system if and only if both eigenvalues are nonzero; .
Reading: §6.2 Linear and Almost Linear Systems.
Video: (Kaltura07)
Friday, September 12th
Topics: Review of stability of linear systems; behaviour of eigenvalues under linear perturbations; stability of linear systems under linear perturbations; stability of linear systems under nonlinear perturbations; stability of nonlinear systems.
Reading: §6.2 Linear and Almost Linear Systems.
Video: (Kaltura08)
Week of September 15th (Week 4)
O5 is due Tuesday, September 16th at 11:00 PM.
O6 is due Friday, September 19th at 11:00 PM.
H3 is due Saturday, September 20th at 11:00 PM.
Monday, September 15th
Topics: Review of last week; predator-prey systems; linearizations of the predator-prey system; phase plane of the the predator-prey system; logistic model; other types of interacting models.
Reading: §6.3 Predator-Prey and Competitions.
Video: (Kaltura09)
Wednesday, September 17th
Topics: Logistic models for single populations; interaction models between two populations (competition, cooperation, predation); in-depth analysis of competition models; the question of co-existence via stability of critical points; phase planes of competition models; parametrix.
Reading: §6.3 Predator-Prey and Competitions.
Video: (Kaltura10)
Friday, September 19th
Topics: Laplace transforms \(\mathscr{L}\{f(t)\} = F(s)\); improper integrals and convergence/divergence; Laplace transforms of \(1\) and \(e^{at}\); Linearity of Laplace transforms; Laplace transforms of \(\sin kt\), \(\cos kt\), and \(t\); existence and uniqueness of Laplace transforms.
Reading: §7.1 Laplace Transforms and Inverse Transforms.
Video: (Kaltura11)
Week of September 22nd (Week 5)
I will be out of town for part of this week; see below for necessary adjustments.
Office Hours will be on M, 11:00 AM to 12:30 PM (no Office Hours on F) this week (in view of my travel).
O7 is due Monday, September 22nd at 11:00 PM.
H4 is due Tuesday, September 30th at 11:00 PM.
Monday, September 22nd
Topics: Tables of Laplace transforms; piece-wise continuous; exponential order; existence and uniqueness of Laplace transforms of functions which are piece-wise continuous and exponential order; inverse Laplace transforms; review of linear \(n\)th order constant coefficient ODEs; Laplace tranforms of derivatives; using Laplace transforms to convert IVPs into algebraic equations; partial fraction decomposition; inverting the Laplace transform via table; a nonhomogenous example.
Reading: §7.2 Transformation of Initial Value Problems.
Video: (Kaltura12)
Wednesday, September 24th
Friday, September 26th
Online Class (via Zoom).
Topics: Review of last class; Laplace transforms of integrals; translations of Laplace transforms; examples of IVPs solved using translations of Laplace transforms.
Reading: §7.2 Transformation of Initial Value Problems; §7.3 Translation of Laplace Transforms.
Notes: (Sept 26th)
Video: (Kaltura13)
Week of September 29th (Week 6)
O8 is due Monday, September 29th at 11:00 PM.
H4 is due Tuesday, September 30th at 11:00 PM.
O9 is due Friday, October 3rd at 11:00 PM.
Monday, September 29th
Topics: Derivatives of Laplace transforms; using these to compute inverse Laplace transforms; higher derivatives of Laplace transforms; integral of the Laplace transform; convolution of functions; convolution is commutative; Laplace transforms take convolution to products.
Reading: §7.4 Derivative, Integral, and Multiplication of Laplace Transforms.
Video: (Kaltura14)
Wednesday, October 1st
Topics: Review of integral of the Laplace transform; review of convolution of functions; review of Laplace transforms take convolution to products; using convolutions to compute inverse Laplace transforms.
Reading: §7.4 Derivative, Integral, and Multiplication of Laplace Transforms.
Video: (Kaltura15)
Friday, October 3rd
Week of October 6th (Week 7)
Midterm 1 is on Monday, October 6th from 6:30 PM to 7:30 PM in MSEE B012.
O10 is due Wednesday, October 8th at 11:00 PM.
O11 is due Friday, October 10th at 11:00 PM.
Monday, October 6th
Topics: Response of a spring system (with friction) to an external force; the (Heaviside) step function \(u(t-a)\); expressing piece-wise continuous using step functions; the Laplace transform \(\mathscr{L}\{u(t-c)\} = e^{-cs}/s\); the Laplace transform \(\mathscr{L}\{u(t-c)f(t-c)\}\); solving IVPs with discontinuous input functions.
Reading: §7.5 Discontinuous Input Functions.
Video: (Kaltura16)
Wednesday, October 8th
Topics: Impulse forces; the (Dirac) delta “function” \(\delta(t-a)\); \(\delta()\) as a limit of step functions; the definition of \(\delta\) as a component of the integrand of an integral; the Laplace transform \(\mathscr\{\delta(t-c)\}\); solving IVPs with impulse inputs; Duhamel's principle.
Reading: §7.6 Impulse Functions.
Video: (Kaltura17)
Friday, October 10th
Topics: Review of Duhamel's principle; solvability (and lack thereof) of first order ODEs; Euler's method for numerical integration of a first order ODE; geometric picture for Euler's method; algorithm for Euler's method; step size \(h\) and update steps for \(x_j\) and \(y_j\); local versus global error; global error of Euler's method is proportional to step size; backward Euler's method; geometric picture for backward Euler's method; implicit update step for \(y_j\); algorithm for backward Euler's method.
Reading: §2.4 Euler's Method; §2.5 Improved Euler Method.
Video: (Kaltura18)
Week of October 13th (Week 8)
Office Hours will be on WF, 11:00 AM to 12:30 PM this week (in view of Fall Break).
O12 is due Wednesday, October 15th at 11:00 PM.
O13 is due Friday, October 17th at 11:00 PM.
H5 is due Saturday, October 18th at 11:00 PM.
Monday, October 13th
Wednesday, October 15th
Friday, October 17th
Week of October 20th (Week 9)
O14 is due Monday, October 20th at 11:00 PM.
O15 is due Friday, October 24th at 11:00 PM.
H6 is due Tuesday, October 21st (note: non-standard day!) at 11:00 PM.
H7 is due Saturday, October 25th at 11:00 PM.
Monday, October 20th
Wednesday, October 22nd
Friday, October 24th
Week of October 27th (Week 10)
O16 is due Monday, October 27th at 11:00 PM.
O17 is due Friday, October 31st at 11:00 PM.
H8 is due Saturday, November 1st at 11:00 PM.
Monday, October 27th
Wednesday, October 29th
Friday, October 31st
Week of November 3rd (Week 11)
O18 is due Monday, November 3rd at 11:00 PM.
O19 is due Friday, November 7th at 11:00 PM.
Monday, November 3rd
Wednesday, November 5th
Friday, November 7th
Week of November 10th (Week 12)
Midterm 2 is on Monday, November 10th from 6:30 PM to 7:30 PM in MSEE B012.
O20 is due Friday, November 14th at 11:00 PM.
H9 is due Saturday, November 15th at 11:00 PM.
Monday, November 10th
Wednesday, November 12th
Friday, November 14th
Week of November 17th (Week 13)
O21 is due Friday, November 21st at 11:00 PM.
H10 is due Saturday, November 22nd at 11:00 PM.
Monday, November 17th
Wednesday, November 19th
Friday, November 21st
Week of November 24th (Week 14)
Monday, November 24th
Wednesday, November 26th
Friday, November 28th
Week of December 1st (Week 15)
I will be out of town this week; see below for necessary adjustments.
There will be a substitute professor teaching all classes in-person in the usual time and place.
Office Hours details TBA.
O22 is due Monday, December 1st at 11:00 PM.
O23 is due Friday, December 5th at 11:00 PM.
H11 is due Saturday, December 6th at 11:00 PM.
Monday, December 1st
Wednesday, December 3rd
Friday, December 5th
Week of December 8th (Week 16)
Monday, December 8th
Wednesday, December 10th
Friday, December 12th
Week of December 15th (Final Exam Week)
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