Concourse Math 18.03 – Calendar of topics and HW assignments  Spring 2020
last updated Tuesday, February 11, 2020 0:02 AM
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  Mon, Feb 3  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) 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 Tues, Feb 11) 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  Tues, Feb 4  [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  Wed, Feb 5  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) 
Numerical Methods: Euler's Method (OCW Video) Example of Euler's Method (OCW Video) 
R2  Thurs, Feb 6  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 Wed, Feb 19) EP 1.5 (Linear 1st Order Equations); IR1–3 (InputResponse Models); SN 2 (Modeling by 1st Order Linear ODEs) SN 3 (Solutions of 1st Order Linear ODEs) 
L3  Mon, Feb 10  Linearity and linear models, continued; Variation of Parameters; higher order linear ODEs; signalresponse perspective; linear system response to exponential and sinusoidal input. Lecture #3 Notes (may be revised) 

R3  Tues, Feb 11  Solutions of first order linear ODEs, integrating factors; Transients; Diffusion example, coupling constant; variation of parameters.  
L4  Wed, Feb 12  Linear system response to exponential and sinusoidal input; gain, phase lag. Lecture #4 Notes (may be revised) 
Problem Set #3 Exam Topics & Practice Questions for Exam #1 Practice Exam 1 Solutions 
R4  Thurs, Feb 13  Complexvalued equation associated to sinusoidal input.  
  Mon, Feb 17  Presidents Day  no classes 

L5  Tues, Feb 18  [Monday schedule] Complexvalued equation associated to sinusoidal input. The algebra of complex numbers; the complex exponential; complex numbers, roots of unity. Complexvalued equation associated to sinusoidal input; gain, phase lag. Lecture #5 Notes (may be revised) 

L6  Wed, Feb 19  Autonomous equations, the phase line, equilibria, critical points, stability. 2nd order linear constant coefficient ODEs, characteristic polynomial. Lecture #67 Notes (may be revised) 

R5  Thurs, Feb 20  Review of topics for exam I  slope fields, integral curves, modeling by ODEs, separable equations, autonomous equations, linear equations, methods.  Problem Set #4 
L7  Mon, Feb 24  Exam #1 [date may change] Solutions  
R6  Tues, Feb 25  Good vibrations, damping conditions [Complex roots; Under, over, critical damping; Complex replacement, extraction of real solutions; Transience; Root diagram]  
L8  Wed, Feb 26  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  Thurs, Feb 27  Linear Algebra: Linear spaces, function spaces and linear operators, span, image and kernel.  
L9  Mon, Mar 2  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  Tues, Mar 3  Dashpot drive; RLC circuits A Worked Example of a 2nd Order Linear System with Sinusoidal Input Signal 

L10  Wed, Mar 4  Linear timeinvariant (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 
R9  Thurs, Mar 5    
L11  Mon, Mar 9  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  Tues, Mar 10  Resonance, Frequency response, LTI systems, superposition, RLC circuits [Resonance; Frequency response; RLC circuits; Time invariance] 

L12  Wed, Mar 11  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  Thurs, Mar 12  Engineering applications [Damping ratio]  Problem Set #6 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  Mon, Mar 16  Fourier series for periodic inputs; orthogonality; Fourier coefficients [Periodic input functions] Lecture #13 Notes (may be revised) 

R12  Tues, Mar 17  Operations on Fourier Series [Squarewave; Piecewise continuity; Tricks: trig id, linear combination, shift]  
L14  Wed, Mar 18  Fourier Series, continued. Fourier's Theorem and Fourier coefficients. Lecture #14 Notes (may be revised) 
Practice Questions for Exam #2 Practice Exam 2 Solutions 
R13  Thurs, Mar 19  Fourier series: harmonic response [Differentiating and integrating Fourier series; Harmonic response; Amplitude and phase expression for Fourier series]  
Spring Break  
L15  Mon, Mar 30  Exam #2 [date may change] Solutions 

R14  Tues, Mar 31  Periodic solutions; resonance; solutions of 1st and 2nd order ODEs with discontinuous input signals; harmonic response, step response.  Problem Set #7 
L16  Wed, Apr 1  Generalized functions, generalized derivative, step and delta functions. Impulse and step responses. Lecture #15 Notes (new, may be revised) 

R15  Thurs, Apr 2  Step and delta functions. Impulse and step responses, generalized derivative.  
L17  Mon, Apr 6  Laplace transform: basic properties (region of convergence; L[t^n]; sshift rule; L[sin(at)] and L(cos(at)]; tdomain vs sdomain; L[delta(t)]; tderivative rule; idea of how to solve ODEs by translating differential equations into algebraic equations. Lecture #16 Notes (new, may be revised) 

R16  Tues, Apr 7  Step and delta functions. Impulse and step responses, generalized derivative. 

L18  Wed, Apr 8  Postinitial conditions of unit impulse response; time invariance; commutation with D; commutation with tshift; Convolution product, delta function as unit for convolution [Worked Examples of Laplace Transform and Convolution] 

R17  Thurs, Apr 9  Solution with initial conditions as w*q. Inverse transform; Partial fractions, coverup; Nonrest initial conditions for first order equations].  Problem Set #8 Exam #3 Topics and Practice Exam #3 (with solutions). This includes a 2page summary of recent topics, a reference sheet for Laplace transforms and Fourier series, a practice exam, and solutions. 
L19  Mon, Apr 13  Solution with of ODEs with initial conditions as w*q; partial fraction methods; nonrest initial conditions for first order equations 

R18  Tues, Apr 14  Second order equations; completing the squares; the pole diagram; weight and transfer function; tshift rule; poles; pole diagram of LT and long term behavior], the transfer function and frequency response [Stability; Transfer and gain]  
L20  Wed, Apr 15  Introduction to vector fields and systems of 1st order ODEs; matrix representation. Here's a website that has a good javabased 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 16  Linear algebra concepts: linear independence, span, basis, coordinates; matrix of a linear transformation relative to a basis; Review for exam III. 
Problem Set #9 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) 
R20  Tues, Apr 21  
L21  Wed, Apr 22  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.  
R21  Thurs, Apr 23  Linear systems and matrices; matrix algebra, row reduction, inverse matrices; eigenvalues, eigenvectors, uncoupling a system (diagonalization), evolution matrices. 

L22  Mon, Apr 27  Exam #3 [date may change] Solutions 

R22  Tues, Apr 28  Solving System of 1st Order Linear Differential Equations, continued. 

L23  Wed, Apr 29  Complex or repeated eigenvalues; qualitative behavior of linear systems; phase plane; tracedeterminant plane, stability; phase portraits, morphing of linear phase portraits. Matrix Methods for Solving Systems of 1st Order Linear Differential Equations 

R23  Thurs, Apr 30  Complex or repeated eigenvalues; Qualitative behavior of linear systems; phase plane [Eigenvalues vs coefficients; Complex eigenvalues; Repeated eigenvalues; Defective, complete; Tracedeterminant plane; Stability]  
L24  Mon, May 4  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 
R24  Tues, May 5  Normal modes and the matrix exponential [Matrix exponential; Uncoupled systems; Exponential law] 

L25  Wed, May 6  Nonlinear systems; linearization near equilibria, Jacobian matrices; the nonlinear pendulum; autonomous systems, predatorprey systems.  
R25  Thurs, May 7  Nonlinear systems; linearization near equilibria, Jacobian matrices; the nonlinear pendulum; autonomous systems, predatorprey systems.  
L26  Mon, May 11  Limitations of the linear: limit cycles and chaos, strange attractors.  
R26  Tues, May 12  Last details.  
Practice Final Exam Problems Solutions 

***  Date TBD  Final Exam  Date and Time TBD in 16160 
References:
FH: FarlowHallMcDillWest, Differential Equations & Linear Algebra, Pearson/PrenticeHall, 2nd Edition
EP: C. Henry Edwards and David E. Penney, Elementary Differential Equations with Boundary Value Problems, PrenticeHall, Sixth Edition.
SN: 18.03 Supplementary Notes, available on the course website.
Notes: 18.03 Notes and Exercises available on the course website.