Calendar | Differential Equations | Mathematics

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.

Course calendar.
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