## MATH 215/255 Sep-Dec 2014 Lecture Summary

 We ek Date Lec- ture Contents 1 0901 Labour Day 0903 L01 outline; §0.2 examples and terminology of differential equations (DEs) 0905 L02 PART I. First order ODEs §1.1 integrals as solutions§1.2 slope fields 2 0908 L03 §1.2 slope fields, unique existence of solutions§1.3 separable equations 0910 L04 §1.3 separable equations continued 0912 L05 §1.4 linear equations and integrating factor 3 0915 L06 (§1.5 is skipped) Braun's §1.9: exact equations 0917 L07 Braun's §1.9: integration factor for exact equation §1.6 autonomous equations 0919 L08 §1.6 phase line analysis for autonomous equations §1.7 numerical methods, Euler method 4 0922 L09 §1.7 examples for Euler method PART II. Second order linear ODEs §2.1 terminology, solution space for homogeneous second order linear ODEs 0924 L10 §2.2 Constant coefficient homogeneous second order linear ODEs: distinct and equal real root cases 0926 L11 §2.2.1-2.2.2 complex numbers and Euler's formula, complex roots case 5 0928 L12 §2.2.2 example, and remark on §2.3 §2.4 mechanical vibrations §2.4.1 some examples 1001 midterm exam 1 1003 L13 §2.4.2 free undamped vibration, and §2.4.3 free damped vibration 6 1006 L14 §2.5 nonhomogeneous equations, the method of undetermined coefficients 1008 L15 §2.5.2 more examples of undetermined coefficients§2.5.3 variation of parameters 1010 L16 Examples of variation of parameters §2.6 forced oscillation 7 1013 Thanksgiving Day 1015 L17 §2.6 forced oscillation continued, Tacoma Narrows Bridge (1940) PART III. Laplace transform §6.1 Laplace transform: definition and examples 1017 L18 More examples of Laplace transform, step functions, properties 8 1020 L19 inverse Laplace transform §6.2 Transforms of derivatives and ODEs 1022 L20 solving ODEs with discontinuous forces, shifting properties 1024 L21 §6.2.5 Laplace transform of integral §6.3 Laplace transform of convolution and its application to ODEs We skip transfer functions (§6.2.4) and integral equations (§6.3.3 and part of §6.2.5) 9 1027 L22 §6.4 Dirac delta function in ODE We skip §6.4.4 on beam bending since it is a boundary value problem 1029 L23 PART IV. Linear systems of ODEs §3.1 systems of nonlinear ODEs, solutions of special cases 1031 L24 (We skip §3.2 which is a review of matrices) §3.3 Linear systems of first order ODEs, solution spaces and general solutions §3.4 eigenvalue method 10 1103 L25 §3.4 eigenvalue method: different real eigenvalues case, complex conjugate eigenvalues case 1105 midterm exam 2 1107 L26 §3.7 eigenvalue method: repeated eigenvalues case 11 1110 L27 §3.5 phase plane analysis 1112 L28 §3.9.1 nonhomogeneous system: methods of undetermined coefficients and variation of parameters, lecture notes (matrix exponential is a special case of variation of parameters; we skip eigenvalue method, §3.6, §3.8, §3.9.2, and §3.9.3) 1114 L29 examples of variation of parameters PART V. Nonlinear systems of ODEs §8.1 autonomous systems, trajectories on phase plane, fixed points 12 1117 L30 §8.1 linearization about fixed points §8.2 definition of stability 1119 L31 §8.2 stability of nonlinear fixed point and its linearization 1121 L32 §8.2 continuedteaching evaluation 13 1124 L33 §8.2.4 conservative systems §8.3.1 pendulum 1126 L34 §8.3.2 predator-prey and competition models (global phase protrait will not be in final exam) 1128 conclusion and review