MIT微分方程课程表.docVIP

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 Eulers 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 Eulers 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