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 | |||
