Concourse Math 18.03 Calendar of topics and HW assignments -- Spring 2019

last updated Friday, May 24, 2019 12:56 PM

The topics and assignments will change as the course proceeds!! Check back frequently.

Seq. Date Topics    [Notes and Supplements] Text sections and homework assignments [Solutions]
L1 Wed, Feb 6

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 (may be revised)

http://math.rice.edu/~dfield/dfpp.html 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.]

Problem Set #1 (due Thurs, Feb 14)
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)
Notes D (Definite Integral Solutions)
Notes G.1 (Graphical & Numerical Methods)
SN 1 (Notation & Language)
EP 1.7 (Population models)
EP 7.1 (Equilibrium solutions and stability)
Isoclines applet
SN 2 (Modeling by 1st Order Linear ODEs)
R1 Thurs, Feb 7 [Linear equations, numerical methods and applets, solution by separation if forcing term is constant, examples and methods.] EP 6.1, 6.2 (Numerical Methods)
Notes G.3 (Graphical & Numerical Methods)
L2 Mon, Feb 11

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 (may be revised)

Numerical Methods: Euler's Method (OCW Video)
Example of Euler's Method (OCW Video)
R2 Tues, Feb 12 Linear equations, homogeneous vs. inhomogeneous solutions, method of undetermined coefficients [Homogeneous equation, null signal, diffusion and Newton's Law of Cooling, coupling constant] Problem Set #2 (due Thurs, Feb 21)
EP 1.5 (Linear 1st Order Equations);
IR1–3 (Input-Response Models);
SN 2 (Modeling by 1st Order Linear ODEs)
SN 3 (Solutions of 1st Order Linear ODEs)
L3 Wed, Feb 13

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)

R3 Thurs, Feb 14 Solutions of first order linear ODEs, integrating factors; Transients; Diffusion example, coupling constant; variation of parameters.
- Mon, Feb 18

Presidents Day - no classes

 
L4 Tues, Feb 19

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

Lecture #4 Notes (may be revised)

Problem Set #3 (due Wed, Mar 6)
EP 1.7 (Population models)
EP 7.1 (Equilibrium solutions and stability)
Notes IR.6 (Input Response Models)
SN 4 (Sinusoidal Solutions)
SN 5 (Algebra of Complex Numbers)
C.1-C.4 (Complex Numbers)
SN 6 (The Complex Exponential)

Exam Topics & Practice Questions for Exam #1
(same username/password as solutions)

Practice Exam 1 Solutions

L5 Wed, Feb 20

Complex-valued equation associated to sinusoidal input. The algebra of complex numbers; the complex exponential; complex numbers, roots of unity. Complex-valued equation associated to sinusoidal input; gain, phase lag.

Lecture #5 Notes (may be revised)

R4 Thurs, Feb 21 Complex-valued equation associated to sinusoidal input.
L6 Mon, Feb 25

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

Lecture #6-7 Notes (may be revised)

R5 Tues, Feb 26 Review of topics for exam I - slope fields, integral curves, modeling by ODEs, separable equations, autonomous equations, linear equations, methods.

Problem Set #4 (due Thurs, Mar 14)
EP 2.1 (2nd Order Linear Equations)
EP 2.2 (General solutions of linear equations)
SN 19 (The Wronskian)
EP 2.3 (Homogeneous equations w/constant coefficients)
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)
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)

Additional Notes



L7 Wed, Feb 27 Exam #1     Solutions
R6 Thurs, Feb 28 Good vibrations, damping conditions [Complex roots; Under, over, critical damping; Complex replacement, extraction of real solutions; Transience; Root diagram]
L8 Mon, Mar 4

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

Lecture #8 Notes (may be revised)

R7 Tues, Mar 5 Linear Algebra: Linear spaces, function spaces and linear operators, span, image and kernel.
L9 Wed, Mar 6

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)

R8 Thurs, Mar 7 Dashpot drive; RLC circuits
A Worked Example of a 2nd Order Linear System with Sinusoidal Input Signal
L10 Mon, Mar 11

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, 2018 to include full proof of RRF)

Problem Set #5 (due Thurs, Mar 21)
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 Cookbook

R9 Tues, Mar 12 --
L11 Wed, Mar 13

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

Lecture #11 Notes (may be revised)

R10 Thurs, Mar 14

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

L12 Mon, Mar 18

Examples of Variation of Parameters and the Exponential Shift Rule; Discontinuous inputs; Introduction to Fourier series for periodic inputs.

Lecture #12 Notes (may be revised)

R11 Tues, Mar 19 Engineering applications [Damping ratio] Problem Set #6 (due Thurs, Apr 11)
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)
L13 Wed, Mar 20

Fourier series for periodic inputs; orthogonality; Fourier coefficients [Periodic input functions]

Lecture #13 Notes (may be revised)

R12 Thurs, Mar 21 Operations on Fourier Series [Squarewave; Piecewise continuity; Tricks: trig id, linear combination, shift]
L14 Mon, Apr 1

Fourier Series, continued. Fourier's Theorem and Fourier coefficients.

Lecture #14 Notes (may be revised)

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

Linear nth Order ODE Cookbook

Practice Exam 2 Solutions

R13 Tues, Apr 2 Fourier series: harmonic response [Differentiating and integrating Fourier series; Harmonic response; Amplitude and phase expression for Fourier series]
L15 Wed, Apr 3

Exam #2     Solutions
You may bring the Linear nth Order ODE Cookbook to the exam.

 
R14 Thurs, Apr 4 Periodic solutions; resonance; solutions of 1st and 2nd order ODEs with discontinuous input signals; harmonic response, step response.

Problem Set #7 (due Thurs, Apr 18)
SN 21 (Steps, impulses and generalized functions)
SN 22 (Generalized functions and differential equations)
SN 23 (Impulse and step responses)
Notes IR (Input response models)
Laplace Transform Facts
EP 4.1 (Laplace transforms and inverse transforms)
EP 4.4 (Derivatives, integrals, and products of transforms)
EP 4.2 (Transformation on initial value problems)
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)

L16 Mon, Apr 8

Generalized functions, generalized derivative, step and delta functions. Impulse and step responses.

Lecture #15 Notes (new, may be revised)

R15 Tues, Apr 9 Step and delta functions. Impulse and step responses, generalized derivative.----
L17 Wed, Apr 10

Laplace transform: basic properties (region of convergence; L[t^n]; s-shift rule; L[sin(at)] and L(cos(at)]; t-domain vs s-domain; L[delta(t)]; t-derivative rule; idea of how to solve ODEs by translating differential equations into algebraic equations.

Lecture #16 Notes (new, may be revised)

R16 Thurs, Apr 11 Step and delta functions. Impulse and step responses, generalized derivative.
L18 Wed, Apr 17

Post-initial conditions of unit impulse response; time invariance; commutation with D; commutation with t-shift; Convolution product, delta function as unit for convolution [Worked Examples of Laplace Transform and Convolution]

Lecture #17 Notes (in preparation - really, I swear!)

R17 Thurs, Apr 18 Solution with initial conditions as w*q. Inverse transform; Partial fractions, coverup; Non-rest initial conditions for first order equations].

Problem Set #8 (due Tues, Apr 30)
SN 25 (Laplace Transform technique: coverup)
SN 26 (The Laplace Transform and generalized functions)
Notes H (Heaviside coverup method)
EP 4.3 (Translation and partial fractions)
Laplace Transform Facts
Notes on Convolution (RW)
FH 8.4 (The Convolution Integral and the Transfer Function)
Notes CG (Convolutions and Green's formula)
SN 24 (Convolution)
SN 27 (The pole diagram and the Laplace Transform)
SN 28 (Amplitude response and the pole diagram)
SN 29 (The Laplace Transform and more general systems)

Exam #3 Topics and Practice Exam #3 (with solutions). This includes a 2-page summary of recent topics, a reference sheet for Laplace transforms and Fourier series, a practice exam, and solutions.

L19 Mon, Apr 22

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

R18 Tues, Apr 23 Second order equations; completing the squares; the pole diagram; weight and transfer function; t-shift rule; poles; pole diagram of LT and long term behavior], the transfer function and frequency response [Stability; Transfer and gain]
L20 Wed, Apr 24

Introduction to vector fields and systems of 1st order ODEs; matrix representation.

Here's a website that has a good java-based tool for showing vector fields and flows: http://math.rice.edu/~dfield/dfpp.html. 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.]

R19 Thurs, Apr 25

Linear algebra concepts: linear independence, span, basis, coordinates; matrix of a linear transformation relative to a basis; Review for exam III.

Problem Set #9 (due Fri, May 10)
EP 5.1 (First-Order systems and applications)
EP 5.2 (The method of elimination)
EP 5.3 (Matrices and linear systems)

Notes LS.1 (Linear systems: Review of linear algebra)

Notes LS.2.2 (Homogeneous linear systems w/constant coefficients)

EP 5.4 (The eigenvalue methods for homogeneous systems)

SN 30 (First order systems and second order equations)

SN 31 (Phase portraits in two dimensions)

Supplement on Evolution Matrices

Supplement on Linear Coordinates, Alternate Bases, and Evolution Matrices

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

Phase portraits for the linear ODE examples

Notes LS.3 (Complex and repeated eigenvalues)

EP 7.2 (Stability and the phase plane)

Notes GS.1–5 (Graphing ODE systems)

EP 5.7 (Matrix exponentials and linear systems)

Notes LS.6 (Solution matrices)

L21 Mon, Apr 29 Vector fields and systems of 1st order ODEs; reduction of order - nth order equations and systems of 1st order equations; Linear systems, matrix definitions and methods; 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; Sampler of Phase Plane diagrams for uncoupled, coupled, period, and stable sink linear systems and a nonlinear system.
R20 Tues, Apr 30

Linear systems and matrices; matrix algebra, row reduction, inverse matrices; eigenvalues, eigenvectors, uncoupling a system (diagonalization), evolution matrices.

L22 Wed, May 1

Exam #3     Solutions
The exam will cover (1) Fourier Series with applications to ODEs with periodic input signals; (2) Generalized functions and generalized derivatives, delta functions, step functions, and box functions; (3) Laplace transform with applications to solving ODEs; (4) convolution of functions, especially convolution of unit impulse response for a differential operator with a given input signal.

R21 Thurs, May 2

Solving System of 1st Order Linear Differential Equations, continued.

L23 Mon, May 6

Complex or repeated eigenvalues; qualitative behavior of linear systems; phase plane; trace-determinant plane, stability; phase portraits, morphing of linear phase portraits.

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

Phase portraits for the linear ODE examples

R22 Tues, May 7 Complex or repeated eigenvalues; Qualitative behavior of linear systems; phase plane [Eigenvalues vs coefficients; Complex eigenvalues; Repeated eigenvalues; Defective, complete; Trace-determinant plane; Stability]
L24 Wed, May 8 Matrix exponentials [Inhomogeneous linear systems (constant input signal)] Problem Set #10 (practice only - solns. will be posted)
EP 7.3 (Linear and almost linear systems)
EP 7.4 (Ecological models: Predators and competitors)
EP 7.5 (Nonlinear mechanical systems)
Notes GS.6 (Graphing ODE systems)
Notes GS.7 (Structural stability)
Nonlinear Systems and Linearization
R23 Thurs, May 9

Normal modes and the matrix exponential [Matrix exponential; Uncoupled systems; Exponential law]
Worked Examples: Matrix Methods for Solving Systems of 1st Order Linear Differential Equations

L25 Mon, May 13 Nonlinear systems; linearization near equilibria, Jacobian matrices; the nonlinear pendulum; autonomous systems, predator-prey systems.
R24 Tues, May 14 Nonlinear systems; linearization near equilibria, Jacobian matrices; the nonlinear pendulum; autonomous systems, predator-prey systems.
L26 Wed, May 15 Limitations of the linear: limit cycles and chaos, strange attractors.
R25 Thurs, May 16 Last details.
   

Practice Final Exam Problems     Solutions

*** Wed, May 22

Final Exam -- Wednesday, May 22 from 1:30pm to 4:30pm in 16-160

References:
FH: Farlow-Hall-McDill-West, Differential Equations & Linear Algebra, Pearson/Prentice-Hall, 2nd Edition
EP: C. Henry Edwards and David E. Penney, Elementary Differential Equations with Boundary Value Problems, Prentice-Hall, Sixth Edition.
SN: 18.03 Supplementary Notes, available on the course website.
Notes: 18.03 Notes and Exercises available on the course website.

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