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 |