Basic notions: Autonomous vs. nonautonomous differential equations, direction fields, isoclines, integral curves, existence and uniqueness of solutions (general solutions, particular solutions with initial conditions), models, numerical/graphical solutions. Linear equations, separable equations (exponential growth with harvesting, mixing problems, cooling problems), system/signal or input/response perspective; separatrices, funnels, other graphical methods. Linear equations, homogeneous vs. inhomogeneous solutions. Solving 1st order linear equations by integrating factors (next time).

Lecture #1 Notes     Sept 4, 2023 Zoom notes     Lecture #2 Notes

https://www.cs.unm.edu/~joel/dfield/ has a good tool for drawing direction (slope) fields. The current version requires you to download a Java executable file to your own computer and to run it locally on your own machine. You can customize various options. You can also print the graphs. [How Java is called varies on computer and operating system, so we may have to provide some additional documentation.] [Also available here]

Problem Set #1 (due by Fri, Sept 13, 11:59pm)
EP 1.1 (Differential equations and mathematical models)
EP 1.2 (Integrals as general and particular solutions)
EP 1.3 (Slope fields and solution curves)
EP 1.4 (Separable equations and applications)
EP 1.5 (Linear 1st Order Equations; integrating factors)
EP 6.1, 6.2 (Numerical Methods)
Numerical Methods: Euler’s Method (OCW Video)
Example of Euler’s Method (OCW Video)

Existence and uniqueness of solutions and Integral Curve Theorem; linearity and linear models - banking, Newton’s Law of Cooling, Hooke’s Law and mass-spring systems with friction and driving force; homogeneous vs. inhomogeneous solutions, method of undetermined coefficients; higher order linear ODEs; input/signal-response perspective; linear system response to exponential and sinusoidal input; Solutions of first order linear ODEs, integrating factors; Transients; Diffusion example.

Lecture #1 Notes     Lecture #2 Notes     Sept 11 Zoom notes

Autonomous equations, the phase line, equilibria, critical points, stability. Analytic solution to logistic equation. Linearity methods - homogeneous, particular, and general solutions to a linear ODE; Undetermined Coefficients, Variation of Parameters, and other methods; input-response perspective; Linear system response to polynomial, exponential and sinusoidal inputs; Exponential Response Formula; gain, phase lag (see notes).

Lecture #3 Notes     Sept 18 Zoom notes

Complex-valued equation associated to sinusoidal input. The algebra of complex numbers; the complex exponential; complex numbers, roots of unity. Applications to trigonometry, integration, and solving ODEs (complex replacement). Complex-valued equation associated to sinusoidal input; gain, phase lag. Exponential Response Formula.

Linear Algebra Interlude: Subspaces, span, image and kernel, linear independence, basis, dimension, coordinates relative to a basis; Linear spaces, function spaces and linear operators, span, image and kernel. (next week)

2nd order linear constant coefficient ODEs, characteristic polynomial, modes, independence of solutions, and superposition of solutions; Wronskian matrix and Wronskian determinant; sinusoidal and exponential response; complex characteristic roots. (partially covered, more next week)

Lecture #4 Notes     Sept 25 Zoom notes

Problem Set #4 (due Fri, Oct 4)
EP 2.1 (2nd Order Linear Equations)
EP 2.2 (General solutions of linear equations)

Subspaces, Span, Linear Independence, Basis of a Subspace; Images and Kernel of a Matrix (RW)

General Linear Spaces (Vector Spaces) and Solutions of ODEs (RW)

EP 2.3 (Homogeneous equations w/constant coefficients)

Exam #1 Topics and Practice Exam Questions
(use same username/password as HW solutions)
the actual exam will likely be shorter and simpler

Practice Exam #1 Solutions

Linear operators with constant coefficients (time invariant), exponential solutions, characteristic polynomial; examples of homogeneous solutions with distinct real roots, pure complex roots (Hooke’s Law); Linear time-invariant (LTI) operators; case of repeated roots of characteristic polynomial; Operators and the Exponential Response Formula (ERF) and the Resonance Response Formula (RRF) for exponential and sinusoidal input signals. Gain and phase lag; spring drive, complex replacement, complex gain, phase lag; Resonance and forced harmonic motion; Dashpot drive; RLC circuits.

A Worked Example of a 2nd Order Linear System with Sinusoidal Input Signal

Lecture #5 Notes     Oct 2 Zoom notes

Exam #1 will take place in Proctorio during a 25 hour window opening Wed. night at 11:00pm (Oct 4) and closing Thursday night at 11:59pm (Oct 5). The exam should take approximately 70 minutes and extra time will be allotted for downloading, printing, scanning, and uploading for a total of 90 minutes. You should take the Proctorio Setup Quiz in Canvas as soon as possible.

Problem Set #5 (due Fri, Oct 11)
EP 2.4 (Mechanical vibrations)
EP 2.5 (Nonhomogeneous equations and undetermined coefficients)
EP 2.6 (Forced oscillations and resonance)
EP 2.7 (Electrical circuits)

Exam #1 solutions

Exponential Response Formula (ERF), Resonance Response Formula (RRF); Resonance, Frequency response, LTI systems, superposition, RLC circuits [Resonance; Frequency response; RLC circuits; Time invariance]; Exponential Shift Rule; Variation of Parameters for higher order systems; Discontinuous inputs. Periodic inputs and introduction to Fourier series methods.

Lecture #6 Notes     Oct 9 Zoom notes

Problem Set #6 (due Fri, Oct 18)

Linear nth Order ODE Cookbook

Fourier series: orthogonality, inner products, orthogonal projection, Pythagorean Theorem; Applications to ODEs - harmonic response, resonance; Fourier series for periodic inputs; Fourier’s Theorem and Fourier coefficients; squarewave function; Sawtooth function; Differentiating and integrating Fourier series; Tips & Tricks: trig id, linear combination, shift

Lecture #7 Notes     Oct 16 Zoom notes

Problem Set #7 (due Fri, Oct 25)
EP 8.1 (Periodic functions and trigonometric series)
EP 8.2 (General Fourier series and convergence)
EP 8.3 (Fourier sine and cosine series)
EP 8.4 (Applications of Fourier series)

Continuation of Fourier series and applications. Generalized functions, generalized derivative, step and delta functions; introduction to the Laplace Transform and how we’ll use it to solve ODEs.

Lecture #8 Notes     Oct 23 Zoom notes     Laplace Transform Facts
These notes contain some material that will be covered in greater depth in next week’s lecture.

Problem Set #8 (due Fri, Nov 1)
EP 8.2 (General Fourier series and convergence)
FH 8.3 (The Step Function and Delta Function)
EP 4.1 (Laplace transforms and inverse transforms)

Generalized functions, generalized derivative, Heaviside, box, and delta functions. Laplace transform: basic properties, rules and sample calculations; t-domain vs s-domain; solving Initial Value Problems (IVPs) by translating differential equations into algebraic equations. Impulse and step responses, unit impulse response; weight function and transfer function. Inverse transform; non-rest initial conditions for first order equations.

Lecture #9 Notes     Oct 30 Zoom notes

Laplace Transform Facts

Problem Set #9 (due Fri, Nov 8)
FH 8.1 (The Laplace Transform and its inverse)
FH 8.2 (Solving DEs and IVPs with Laplace Transforms)
FH 8.3 (The Step Function and Delta Function)
EP 4.1 (Laplace transforms and inverse transforms)
EP 4.2 (Transformation of initial value problems)
EP 4.3 (Translation and partial fractions)
EP 4.4 (Derivatives, integrals, and products of transforms)
FH 8.4 (The Convolution Integral and the Transfer Function)

Laplace Transform Facts

Worked Examples of Laplace Transform and Convolution

Notes on Convolution (RW)

Zero State Response and Zero Input Response (ZSR + ZIR); Convolution Integral to determine Zero State Response (ZSR) using weight function from unit impulse response; solution with initial conditions as w*q.

Introduction to vector fields and systems of 1st order ODEs; reduction of order - nth order equations and systems of 1st order equations.

Lecture #10 Notes     Nov 6 Zoom notes

Notes on Continuous Dynamical Systems - Part 1

Here’s a website that has a good java-based tool for showing vector fields and flows: https://www.cs.unm.edu/~joel/dfield/ The current version requires you to download a Java executable file to your own computer and to run it locally on your own machine. You can customize various options. You can also print the graphs. [How Java is called varies on computer and operating system, so we may have to provide some additional documentation.] [Also available here]

Problem Set #10 (due Fri, Nov 15)
EP 4.4 (Derivatives, integrals, and products of transforms)
FH 8.4 (The Convolution Integral and the Transfer Function)

Worked Examples of Laplace Transform and Convolution

Notes on Convolution (RW)

EP 5.1 (First-Order systems and applications)
EP 5.2 (The method of elimination)

Phase portraits for the linear ODE examples (RW)

The 2nd Midterm exam will take place during the 11th week of the course - again in Proctorio. The main topics will be some of the later topics in solving higher-order linear ODEs, periodically driven systems (Fourier Series), and Laplace Transform methods. The problem sets should be a good indication, but algebraically more straightforward. A Table of Fourier Series Facts and Laplace Transform Facts will be provided with the exam.

The exam window will open at 11:00pm EST on Wed, Nov 13 and will close at 11:59pm EST on Thurs, Nov 14.

Exam #2 Topics, Study Guide, and Practice Exam Questions     Solutions

Wed, Nov 13
Lecture #11

First order linear systems of ODEs in matrix form, solution of uncoupled (diagonal) systems and evolution matrices; uncoupling a system (diagonalization) in case of real eigenvalues, evolution matrices; Phase Plane diagrams for uncoupled systems.

Linear algebra basics: linear independence, span, basis, coordinates; matrix of a linear transformation relative to a basis. Matrix of a linear transformation relative to a basis and linear coordinate changes to standardize systems of 1st order linear equations. Solving systems of 1st Order Linear Differential Equations; real eigenvalue case; matrix methods for solving systems of 1st order linear differential equations; phase portraits for the linear ODE examples.

Lecture #11 Notes     Nov 13 Zoom notes

Notes on Coordinate Changes (general idea)

Notes on Continuous Dynamical Systems - Part 1

Notes on Continuous Dynamical Systems - Part 2

Exam #2 solutions

Problem Set #11 (due Fri, Nov 22)
EP 7.2 (Stability and the phase plane)
EP 5.3 (Matrices and linear systems)
EP 5.4 (The eigenvalue methods for homogeneous systems)
EP 5.7 (Matrix exponentials and linear systems)

Supplement on Evolution Matrices (RW)

Supplement on Linear Coordinates, Alternate Bases, and Evolution Matrices (RW)

Matrix Methods for Solving Systems of 1st Order Linear Differential Equations (RW)

Supplement on Representation of Functions in Different Coordinates

Solving systems of 1st Order Linear Differential Equations; real eigenvalue case; complex eigenvalue case; repeated eigenvalue case. Qualitative behavior of linear systems; phase plane; matrix methods for solving systems of 1st order linear differential equations; phase portraits for the linear ODE examples. Sampler of Phase Plane diagrams for uncoupled, coupled, period, and stable sink linear systems and a nonlinear system.

Lecture #12 Notes     Nov 20 Zoom notes

Notes on Continuous Dynamical Systems - Part 2

Lecture #13 Notes may also be useful.

Problem Set #12 (due Fri, Dec 6)
EP 5.4 (The eigenvalue methods for homogeneous systems)
EP 5.7 (Matrix exponentials and linear systems)

Notes on Coordinate Changes (general idea)

Supplement on Representation of Functions in Different Coordinates

Supplement on Evolution Matrices

Matrix Methods for Solving Systems of 1st Order Linear Differential Equations (RW)

Solving systems of 1st Order Linear Differential Equations; complex eigenvalue case; repeated eigenvalue case. Decomposition of 1st order linear system into mode (block matrices). Qualitative behavior of linear systems. Fundamental matrices (see notes). Simple nonlinear systems (a) with shifted equilibrium, and (b) solution using Undetermined Coefficients or Variation of Parameters (see notes).

Lecture #13 Notes     Dec 4 Zoom notes (w/added details)

Notes on Continuous Dynamical Systems - Part 2

Nonlinear systems; linearization near equilibria, Jacobian matrices; the nonlinear pendulum; autonomous systems, predator-prey systems; simple examples of nonautonomous systems.

Lecture #13 Notes     Lecture #14 Notes     Dec 11 Zoom notes

Lorenz System, Lorenz Attractor (Wikipedia)

Mathematical Theory of Epidemics (Kermack, McKendrick, 1927) - 22 pages, somewhat technical, see pgs 713-714 in particular

Problem Set #14. These problems should be done for practice but not turned in. Nonlinear systems and phase plane analysis will appear on the Final Exam. [Solutions will be posted.]

Nonlinear Systems and Linearization

The two-hour Final Exam will take place via Proctorio during a 24-hour window on Dec 18.
The exam window will open at 12:00am EST on Wed, Dec 18 and will close at 11:59pm EST on Wed, Dec 18.

Final Exam Topics and Sampler of Final Exam Problems     Solutions will be posted.
[There will be no regular lecture during the week of the Final Exam.]