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 continued
teaching 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