# 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 # TOPICS SKILLS & CONCEPTS INTRODUCED KEY DATES
I. First-order differential equations
R1 Natural 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

L1 Direction fields, existence and uniqueness of solutions

Direction fields

Integral curve

Isoclines

Funnels

Implicit solutions

Failure of solutions to continue: infinite derivative

PS 1 out
R2 Direction fields, integral curves, isoclines, separatrices, funnels

Separatrix

Extrema of solutions

L2 Numerical methods Euler's method
L3 Linear equations, models

First order linear equation

System/signal perspective

Bank account model

RC circuit

Solution by separation if forcing term is constant

R3 Euler's method; linear models Mixing problems
L4 Solution of linear equations, integrating factors

Homogeneous equation, null signal

Integrating factors

Transients

Diffusion example; coupling constant

R4 First order linear ODEs; integrating factors Sinusoidal input signal
L5 Complex numbers, roots of unity

Complex numbers

Roots of unity

PS 1 due; PS 2 out
L6 Complex exponentials; sinusoidal functions

Complex exponential

Sinusoidal functions: Amplitude, Circular frequency, Phase lag

L7 Linear 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

R5 Complex numbers; complex exponentials
L8 Autonomous equations; the phase line, stability

Autonomous equation

Phase line

Stability

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

PS 2 due; PS 3 out
L9 Linear vs. nonlinear

Non-continuation of solutions

R6 Review for exam I
Exam I    Hour exam I
II. Second-order linear equations
R7 Solutions to second order ODEs

Harmonic oscillator

Initial conditions

Superposition in homogeneous case

L11 Modes and the characteristic polynomial

Spring/mass/dashpot system

General second order linear equation

Characteristic polynomial

Solution in real root case

L12 Good vibrations, damping conditions

Complex roots

Under, over, critical damping

Complex replacement, extraction of real solutions

Transience

Root diagram

R8 Homogeneous 2nd order linear constant coefficient equations

General sinusoidal response

Normalized solutions

L13 Exponential response formula, spring drive

Driven systems

Superposition

Exponential response formula

Complex replacement

Sinusoidal response to sinusoidal signal

R9 Exponential and sinusoidal input signals
L14 Complex gain, dashpot drive

Gain, phase lag

Complex gain

PS 3 due; PS 4 out
L15 Operators, undetermined coefficients, resonance

Operators

Resonance

Undetermined coefficients

R10 Gain and phase lag; resonance; undetermined coefficients
L16 Frequency response Frequency response
R11 Frequency response First order frequency response
L17 LTI systems, superposition, RLC circuits.

RLC circuits

Time invariance

PS4 due; PS 5 out
L18 Engineering applications Damping ratio
R12 Review for exam II
L19 Exam II    Hour Exam II
III. Fourier series
R13 Fourier series: introduction Periodic functions
L20 Fourier series

Fourier series

Orthogonality

Fourier integral

L21 Operations on fourier series

Squarewave

Piecewise continuity

Tricks: trig id, linear combination, shift

R14 Fourier series Different periods
L22 Periodic solutions; resonance

Differentiating and integrating fourier series

Harmonic response

Amplitude and phase expression for Fourier series

R15 Fourier series: harmonic response
L23 Step functions and delta functions

Step function

Delta function

Regular and singularity functions

Generalized function

Generalized derivative

PS 5 due; PS 6 out
L24 Step response, impulse response

Unit and step responses

Rest initial conditions

First and second order unit step or unit impulse response

R16 Step and delta functions, and step and delta responses
L25 Convolution

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

R17 Convolution Delta function as unit for convolution
L26 Laplace 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
L27 Application to ODEs

L[delta(t)]

t-derivative rule

Inverse transform

Partial fractions; coverup

Non-rest initial conditions for first order equations

R18 Laplace transform Unit step response using Laplace transform.
L28 Second order equations; completing the squares

s-derivative rule

Second order equations

R19 Laplace transform II
L29 The 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
L30 The transfer function and frequency response

Stability

Transfer and gain

R20 Review for exam III
Exam III    Hour Exam III
IV. First order systems
L32 Linear systems and matrices

First order linear systems

Elimination

Matrices

Anti-elimination: Companion matrix

R21 First order linear systems
L33 Eigenvalues, eigenvectors

Determinant

Eigenvalue

Eigenvector

Initial values

R22 Eigenvalues and eigenvectors Solutions vs trajectories
L34 Complex or repeated eigenvalues

Eigenvalues vs coefficients

Complex eigenvalues

Repeated eigenvalues

Defective, complete

PS 8 due; PS 9 out
L35 Qualitative behavior of linear systems; phase plane

Trace-determinant plane

Stability

R23 Linear phase portraits Morphing of linear phase portraits
L36 Normal modes and the matrix exponential

Matrix exponential

Uncoupled systems

Exponential law

R24 Matrix exponentials Inhomogeneous linear systems (constant input signal)
L37 Nonlinear systems

Nonlinear autonomous systems

Vector fields

Phase portrait

Equilibria

Linearization around equilibrium

Jacobian matrices

PS 9 due
L38 Linearization near equilibria; the nonlinear pendulum

Nonlinear pendulum

Phugoid oscillation

Tacoma Narrows Bridge

R25 Autonomous systems Predator-prey systems
L39 Limitations of the linear: limit cycles and chaos

Structural stability

Limit cycles

Strange attractors

R26 Reviews
Final exam