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SES #TOPICSSKILLS & CONCEPTS INTRODUCEDKEY DATESI. First-order differential equationsR1Natural growth, separable equationsModeling: exponential growth with harvestingGrowth rateSeparating variablesSolutions, general and particularAmalgamating constants of integrationUse of ln|y|, and its eliminationReintroduction of lost solutionsInitial conditions - satisfying them by choice of integration constant L1Direction fields, existence and uniqueness of solutionsDirection fieldsIntegral curveIsoclinesFunnelsImplicit solutionsFailure of solutions to continue: infinite derivativePS 1 outR2Direction fields, integral curves, isoclines, separatrices, funnelsSeparatrixExtrema of solutions L2Numerical methodsEulers method L3Linear equations, modelsFirst order linear equationSystem/signal perspectiveBank account modelRC circuitSolution by separation if forcing term is constant R3Eulers method; linear modelsMixing problems L4Solution of linear equations, integrating factorsHomogeneous equation, null signalIntegrating factorsTransientsDiffusion example; coupling constant R4First order linear ODEs; integrating factorsSinusoidal input signal L5Complex numbers, roots of unityComplex numbersRoots of unityPS 1 due; PS 2 outL6Complex exponentials; sinusoidal functionsComplex exponentialSinusoidal functions: Amplitude, Circular frequency, Phase lag L7Linear system response to exponential and sinusoidal input; gain, phase lagFirst order linear response to exponential or sinusoidal signalComplex-valued equation associated to sinusoidal inputPS: half life R5Complex numbers; complex exponentials L8Autonomous equations; the phase line, stabilityAutonomous equationPhase lineStabilityek(t-t_0) vs cektPS 2 due; PS 3 outL9Linear vs. nonlinearNon-continuation of solutions R6Review for exam I Exam I Hour exam III. Second-order linear equationsR7Solutions to second order ODEsHarmonic oscillatorInitial conditionsSuperposition in homogeneous case L11Modes and the characteristic polynomialSpring/mass/dashpot systemGeneral second order linear equationCharacteristic polynomialSolution in real root case L12Good vibrations, damping conditionsComplex rootsUnder, over, critical dampingComplex replacement, extraction of real solutionsTransienceRoot diagram R8Homogeneous 2nd order linear constant coefficient equationsGeneral sinusoidal responseNormalized solutions L13Exponential response formula, spring driveDriven systemsSuperpositionExponential response formulaComplex replacementSinusoidal response to sinusoidal signal R9Exponential and sinusoidal input signals L14Complex gain, dashpot driveGain, phase lagComplex gainPS 3 due; PS 4 outL15Operators, undetermined coefficients, resonanceOperatorsResonanceUndetermined coefficients R10Gain and phase lag; resonance; undetermined coefficients L16Frequency responseFrequency response R11Frequency responseFirst order frequency response L17LTI systems, superposition, RLC circuits.RLC circuitsTime invariancePS4 due; PS 5 outL18Engineering applicationsDamping ratio R12Review for exam II L19Exam II Hour Exam IIIII. Fourier seriesR13Fourier series: introductionPeriodic functions L20Fourier seriesFourier seriesOrthogonalityFourier integral L21Operations on fourier seriesSquarewavePiecewise continuityTricks: trig id, linear combination, shift R14Fourier seriesDifferent periods L22Periodic solutions; resonanceDifferentiating and integrating fourier seriesHarmonic responseAmplitude and phase expression for Fourier series R15Fourier series: harmonic response L23Step functions and delta functionsStep functionDelta functionRegular and singularity functionsGeneralized functionGeneralized derivativePS 5 due; PS 6 outL24Step response, impulse responseUnit and step responsesRest initial conditionsFirst and second order unit step or unit impulse response R16Step and delta functions, and step and delta responses L25ConvolutionPost-initial conditions of unit impulse responseTime invariance: Commutation with DTime invariance: Commutation with t-shiftConvolution productSolution with initial conditions as w * q R17ConvolutionDelta function as unit for convolution L26Laplace transform: basic propertiesLaplace transformRegion of convergenceLtns-shift ruleLsin(at) and L(cos(at)t-domain vs s-domainPS 6 due; PS 7 outL27Application to ODEsLdelta(t)t-derivative ruleInverse transformPartial fractions; coverupNon-rest initial conditions for first order equations R18Laplace transformUnit step response using Laplace transform. L28Second order equations; completing the squaress-derivative ruleSecond order equations R19Laplace transform II L29The pole diagramWeight and transfer functionLweight function = transfer functiont-shift rulePolesPole diagram of LT and long term behaviorPS 7 due; PS 8 outL30The transfer function and frequency responseStabilityTransfer and gain R20Review for exam III Exam III Hour Exam IIIIV. First order systemsL32Linear systems and matricesFirst order linear systemsEliminationMatricesAnti-elimination: Companion matrix R21First order linear systems L33Eigenvalues, eigenvectorsDeterminantEigenvalueEigenvectorInitial values R22Eigenvalues and eigenvectorsSolutions vs trajectories L34Complex or repeated eigenvaluesEigenvalues vs coefficientsComplex eigenvaluesRepeated eigenvaluesDefective, completePS 8 due; PS 9 outL35Qualitative behavior of linear systems; phase planeTrace-determinant planeStability R23Linear phase portraitsMorphing of linear phase portraits L36Normal modes and the matrix exponentialMatrix exponentialUncoupled systemsExponential law R24Matrix exponentialsInhomogeneous linear systems (constant input signal) L37Nonlinear systemsNonlinear autonomous systemsVector fieldsPhase portraitEquilibriaLinearization around equilibriumJacobian matricesPS 9 dueL38Linearization near equilibria; the nonlinear pendulumNonlinear pendulumPhugoid oscillationTacoma Narrows Bridge R25Autonomous systemsPredator-prey systems L39Limitations of the linear: limit cycles and chaosStructural stabilityLimit cyclesStrange attractors R26Reviews Final exam
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