# Calendar

The calendar below provides information on the course's lecture (L) and recitation (R) sessions. There is also a list of skills and concepts and where they are first introduced. Problem Set (PS) distribution and due dates are also provided.

SES #TOPICSSKILLS & CONCEPTS INTRODUCEDKEY DATES
I. First-order differential equations
R1Natural growth, separable equations

Modeling: exponential growth with harvesting

Growth rate

Separating variables

Solutions, general and particular

Amalgamating constants of integration

Use of ln|y|, and its elimination

Reintroduction of lost solutions

Initial conditions - satisfying them by choice of integration constant

L1Direction fields, existence and uniqueness of solutions

Direction fields

Integral curve

Isoclines

Funnels

Implicit solutions

Failure of solutions to continue: infinite derivative

PS 1 out
R2Direction fields, integral curves, isoclines, separatrices, funnels

Separatrix

Extrema of solutions

L2Numerical methodsEuler's method
L3Linear equations, models

First order linear equation

System/signal perspective

Bank account model

RC circuit

Solution by separation if forcing term is constant

R3Euler's method; linear modelsMixing problems
L4Solution of linear equations, integrating factors

Homogeneous equation, null signal

Integrating factors

Transients

Diffusion example; coupling constant

R4First order linear ODEs; integrating factorsSinusoidal input signal
L5Complex numbers, roots of unity

Complex numbers

Roots of unity

PS 1 due; PS 2 out
L6Complex exponentials; sinusoidal functions

Complex exponential

Sinusoidal functions: Amplitude, Circular frequency, Phase lag

L7Linear system response to exponential and sinusoidal input; gain, phase lag

First order linear response to exponential or sinusoidal signal

Complex-valued equation associated to sinusoidal input

PS: half life

R5Complex numbers; complex exponentials
L8Autonomous equations; the phase line, stability

Autonomous equation

Phase line

Stability

e^{k(t-t_0)} vs ce^{kt}

PS 2 due; PS 3 out
L9Linear vs. nonlinear

Non-continuation of solutions

R6Review for exam I
Exam I  Hour exam I
II. Second-order linear equations
R7Solutions to second order ODEs

Harmonic oscillator

Initial conditions

Superposition in homogeneous case

L11Modes and the characteristic polynomial

Spring/mass/dashpot system

General second order linear equation

Characteristic polynomial

Solution in real root case

L12Good vibrations, damping conditions

Complex roots

Under, over, critical damping

Complex replacement, extraction of real solutions

Transience

Root diagram

R8Homogeneous 2nd order linear constant coefficient equations

General sinusoidal response

Normalized solutions

L13Exponential response formula, spring drive

Driven systems

Superposition

Exponential response formula

Complex replacement

Sinusoidal response to sinusoidal signal

R9Exponential and sinusoidal input signals
L14Complex gain, dashpot drive

Gain, phase lag

Complex gain

PS 3 due; PS 4 out
L15Operators, undetermined coefficients, resonance

Operators

Resonance

Undetermined coefficients

R10Gain and phase lag; resonance; undetermined coefficients
L16Frequency responseFrequency response
R11Frequency responseFirst order frequency response
L17LTI systems, superposition, RLC circuits.

RLC circuits

Time invariance

PS4 due; PS 5 out
L18Engineering applicationsDamping ratio
R12Review for exam II
L19Exam II  Hour Exam II
III. Fourier series
R13Fourier series: introductionPeriodic functions
L20Fourier series

Fourier series

Orthogonality

Fourier integral

L21Operations on fourier series

Squarewave

Piecewise continuity

Tricks: trig id, linear combination, shift

R14Fourier seriesDifferent periods
L22Periodic solutions; resonance

Differentiating and integrating fourier series

Harmonic response

Amplitude and phase expression for Fourier series

R15Fourier series: harmonic response
L23Step functions and delta functions

Step function

Delta function

Regular and singularity functions

Generalized function

Generalized derivative

PS 5 due; PS 6 out
L24Step response, impulse response

Unit and step responses

Rest initial conditions

First and second order unit step or unit impulse response

R16Step and delta functions, and step and delta responses
L25Convolution

Post-initial conditions of unit impulse response

Time invariance: Commutation with D

Time invariance: Commutation with t-shift

Convolution product

Solution with initial conditions as w * q

R17ConvolutionDelta function as unit for convolution
L26Laplace transform: basic properties

Laplace transform

Region of convergence

L[t^n]

s-shift rule

L[sin(at)] and L(cos(at)]

t-domain vs s-domain

PS 6 due; PS 7 out
L27Application to ODEs

L[delta(t)]

t-derivative rule

Inverse transform

Partial fractions; coverup

Non-rest initial conditions for first order equations

R18Laplace transformUnit step response using Laplace transform.
L28Second order equations; completing the squares

s-derivative rule

Second order equations

R19Laplace transform II
L29The pole diagram

Weight and transfer function

L[weight function] = transfer function

t-shift rule

Poles

Pole diagram of LT and long term behavior

PS 7 due; PS 8 out
L30The transfer function and frequency response

Stability

Transfer and gain

R20Review for exam III
Exam III  Hour Exam III
IV. First order systems
L32Linear systems and matrices

First order linear systems

Elimination

Matrices

Anti-elimination: Companion matrix

R21First order linear systems
L33Eigenvalues, eigenvectors

Determinant

Eigenvalue

Eigenvector

Initial values

L34Complex or repeated eigenvalues

Eigenvalues vs coefficients

Complex eigenvalues

Repeated eigenvalues

Defective, complete

PS 8 due; PS 9 out
L35Qualitative behavior of linear systems; phase plane

Trace-determinant plane

Stability

R23Linear phase portraitsMorphing of linear phase portraits
L36Normal modes and the matrix exponential

Matrix exponential

Uncoupled systems

Exponential law

R24Matrix exponentialsInhomogeneous linear systems (constant input signal)
L37Nonlinear systems

Nonlinear autonomous systems

Vector fields

Phase portrait

Equilibria

Linearization around equilibrium

Jacobian matrices

PS 9 due
L38Linearization near equilibria; the nonlinear pendulum

Nonlinear pendulum

Phugoid oscillation

Tacoma Narrows Bridge

R25Autonomous systemsPredator-prey systems
L39Limitations of the linear: limit cycles and chaos

Structural stability

Limit cycles

Strange attractors

R26Reviews
Final exam