Basic notions: Autonomous differential equations, direction fields, integral curves, existence and uniqueness of solutions (general solutions, particular solutions with initial conditions), examples, models, numerical/graphical solutions. Linear equations, separable equations (exponential growth with harvesting, mixing problems, cooling problems), system/signal perspective.

Lecture #1 Notes (revised Feb 2, 2020)

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]

Direction fields, integral curves, isoclines, separatrices, funnels, graphical methods. Linear equations, homogeneous vs. inhomogeneous solutions, method of undetermined coefficients. Solving 1st order linear equations by integrating factors and by linearity.

Lecture #2 Notes (revised Feb 5, 2020)

Linearity and linear models, continued; Variation of Parameters; higher order linear ODEs; signal-response perspective; linear system response to exponential and sinusoidal input.

Lecture #3 Notes (may be revised)

Linear system response to exponential and sinusoidal input; gain, phase lag.

Lecture #4 Notes (may be revised)

Presidents Day - no classes

[Monday schedule] 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.

Lecture #5 Notes (revised Feb 19, 2020)

Autonomous equations, the phase line, equilibria, critical points, stability. 2nd order linear constant coefficient ODEs, characteristic polynomial.

Lecture #6-7 Notes (may be revised)

Problem Set #4 (due Tues, Mar 10)
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)
SN 19 (The Wronskian)
EP 2.4 (Mechanical vibrations)
SN 7 (Beats)
SN 8 (RLC circuits)
SN 9 (Normalization of solutions)
SN 10 (Operators and the exponential response formula)
EP 2.5 (Nonhomogeneous equations and undetermined coefficients)

Additional Notes

Linear Algebra: Subspaces, span, image and kernel, linear independence, basis, dimension, coordinates relative to a basis.

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; normalized solutions; harmonic oscillator. Complex characteristic roots. [some topics may be shifted to other lectures]

Lecture #8 Notes (revised Feb 26, 2020)

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).

Lecture #9 Notes (to be revised)

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.

Lecture #10 Notes (revised Mar 17, 2020 to include full proof of RRF)

Problem Set #5 (due Wed, Mar 18)
SN 10 (Operators and the exponential response formula)
EP 2.6 (Forced oscillations and resonance)
SN 12 (Resonance)
Notes O (Linear differential operators)
SN 11 (Undetermined coefficients)
SN 13 (Time invariance)
SN 14 (The exponential shift law)
SN 15 (Natural frequency and damping ratio)
SN 16 (Frequency response)
SN 17 (Resonance, not: the Tacomah Narrows Bridge)
EP 2.7 (Electrical circuits)
Review class notes on Exponential Shift Formula and Variation of Parameters.

Linear nth Order ODE Summary

Exponential Response Formula (ERF), Resonance Response Formula (RRF); Exponential Shift Rule; Variation of Parameters for higher order systems.

Lecture #11 Notes (may be revised)

Resonance, Frequency response, LTI systems, superposition, RLC circuits [Resonance; Frequency response; RLC circuits; Time invariance]

Examples of Variation of Parameters and the Exponential Shift Rule; Discontinuous inputs.

Lecture #12 Notes (revised Mar 27, 2020)

Summary of methods for linear ODEs, homogeneous, particular solutions, and linearity principles; Fourier series for periodic inputs; Fourier's Theorem and Fourier coefficients; squarewave function.
Note: The current plan is to do more of a "survey of Fourier Series w/applications to ODEs" rather than the full treatment. You may, if you wish, read things in greater detail in the previously produced Lecture Notes.

This week's Zoom Presentation Notes:

Linear nth Order Summary     Fourier I

Fourier II     Fourier III

Problem Set #6 (due Thurs, Apr 9)
EP 8.1 (Periodic functions and trigonometric series)
SN 20 (More on Fourier series)
EP 8.2 (General Fourier series and convergence)
EP 8.3 (Fourier sine and cosine series)
EP 8.4 (Applications of Fourier series)

Lecture #13 Notes (may be revised)

Lecture #14 Notes (revised Mar 27, 2020)

Sketches used in Mar 31 recitation classes: pg1   pg2   pg3   pg4

Practice Questions and Solutions for Exam #2
(same username/password as solutions)

Fourier series: orthogonality, inner products, orthogonal projection, Pythagorean Theorem; Applications to ODEs - harmonic response, resonance

Sawtooth function; Differentiating and integrating Fourier series; Tips & Tricks: trig id, linear combination, shift

Sketchs used in Apr 2 recitation classes (4 page PDF)

Exam #2    Solutions

Generalized functions, generalized derivative, step and delta functions. Impulse and step responses.
Note: The current plan is to do more of a "survey of Generalized Functions and Laplace Transforms w/applications to ODEs" rather than the full treatment. The goal is to simply introduce you the ideas and illustrate them with a few examples. You may, if you wish, read things in greater detail in the Lecture Notes and in the following Zoom presentation notes:

Lecture #15 Notes (new, may be revised)

Delta-Laplace 1     Delta-Laplace 2     Delta-Laplace 3

Step and delta functions. Impulse and step responses, generalized derivative; unit impulse response; time invariance; Convolution product.

Lecture #17-18 Notes

Sketches used in Apr 9 recitation class (PDF)

Solution with initial conditions as w*q. Inverse transform; non-rest initial conditions for first order equations]. [Worked Examples of Laplace Transform and Convolution]; Introduction to systems of 1st order ordinary differential equations and associated vector fields.

Lecture #17-18 Notes

Solution with of ODEs with initial conditions as w*q; partial fraction methods; non-rest initial conditions for first order equations

Sketches used in Apr 14 recitation classes (PDF w/corrections)

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

Sketches used in Apr 15 Lecture (PDF w/additions/corrections)

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]

Sketches used in Apr 16 recitation classes (PDF w/additions/corrections)

Linear algebra: linear independence, span, basis, coordinates; matrix of a linear transformation relative to a basis.

Notes on Coordinate Changes (general idea)

Sketches used in Apr 21 recitation classes (PDF)

Sketches used in Apr 23 recitation classes

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

Phase portraits for the linear ODE examples

Solving System of 1st Order Linear Differential Equations, continued; repeated eigenvalues; decomposition of 1st order linear system into mode (block matrices); simple nonlinear system with shifted equilibrium

Sketches used in Apr 29 Lecture (PDF)

Qualitative behavior of linear systems; phase plane [Eigenvalues vs coefficients; Complex eigenvalues; Repeated eigenvalues; Defective, complete; Trace-determinant plane; Stability]; simple nonlinear systems.

Sketches used in May 4 Lecture (PDF)

p-set and other questions

Sketches used in May 6 Lecture (PDF)

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

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

Practice Final Exam Problems     Solutions

Final Exam -- 9:00am to noon