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. Firstorder 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 lny, 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 Complexvalued 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(tt_0)} vs ce^{kt} 
PS 2 due; PS 3 out 
L9  Linear vs. nonlinear 
Noncontinuation of solutions 

R6  Review for exam I  
Exam I  Hour exam I  
II. Secondorder 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 
Postinitial conditions of unit impulse response Time invariance: Commutation with D Time invariance: Commutation with tshift 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] sshift rule L[sin(at)] and L(cos(at)] tdomain vs sdomain 
PS 6 due; PS 7 out 
L27  Application to ODEs 
L[delta(t)] tderivative rule Inverse transform Partial fractions; coverup Nonrest initial conditions for first order equations 

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

R19  Laplace transform II  
L29  The pole diagram 
Weight and transfer function L[weight function] = transfer function tshift 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 Antielimination: 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 
Tracedeterminant 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  Predatorprey systems  
L39  Limitations of the linear: limit cycles and chaos 
Structural stability Limit cycles Strange attractors 

R26  Reviews  
Final exam 