Math 204, Section 2, Spring 2014

Instructor: Ali Mostafazadeh

Topics Covered in Each Lecture

Lecture Number	Date	Content	Corresponding Reading material
1	Feb. 03	Introduction to and classification of differential equations: ODEs, PDEs, order of a differential equation, linear and  nonlinear equations	Chapter 1 of Boyce-DiPrima, 10th Edition
2	Feb. 05	Definition of a solution of an ODE in an interval, Linear First order ODEs: Method of integrating factor, examples	Page 22 & Section 2.1 of Boyce-DiPrima, 10th Edition
3	Feb. 10	Existence and uniqueness theorem for the solution of the first order linear ODEs, Separable equations, examples	Pages 68-71 & Section 2.2 of Boyce-DiPrima, 10th Edition
4	Feb. 12	Bernouli's Equation, Exact Equations: Definition and a characterization theorem for exact equations (exactness test), examples	Pages 77-78 & 95-99 of Boyce-DiPrima, 10th Edition
5	Feb. 17	Integrating Factors (turning a first order equation into an exact equation); a linear first order initial-value problem with no solutions, Existence and Uniqueness Theorem for first order equations, examples of first order initial-value problems with one, two, or infinitely many solutions	Pages 99-100 & 70-72 of Boyce-DiPrima, 10th Edition
6	Feb. 19	2nd order ODEs, Linear 2nd order ODEs, homogeneous and non-homogeneous equations, function spaces (C^n) as vector spaces, linear maps (operators), the derivative operator D, differential operators, writing a linear ODE as  L y =g where L is a differential operator and g is a given function, solution space of homogeneous linear ODEs, L y=0, as the null space of L, linear equations with constant coefficients, solving 2nd order linear homogeneous ODEs with constant coefficient by reduction to first order linear equations, the characteristic equation	Pages 137-140 of Boyce-DiPrima, 10th Edition
7	Feb. 24	Examples of 2nd order linear homogeneous ODEs with constant coefficient (real and repeated roots), complex numbers, exponential of a complex number and Euler's formula, 2nd order linear homogeneous ODEs with constant coefficient when the characteristic polynomial has complex roots; Existence and uniqueness theorem for 2nd order linear  ODEs (without proof),  Wronskian, the general form of the solution of 2nd order linear homogeneous ODEs.	Pages 158-164 & 145-150 of Boyce-DiPrima, 10th Edition
8	Feb. 26	Abel's theorem and its consequences, Fundamental set of solutions; Theorem: Two solutions of a 2nd order linear homogeneous ODE with continuous coefficients are linearly-dependent if and only if their Wronskian vanishes; the general form of the solution for a 2nd order linear homogeneous ODE with continuous coefficients; the method of reduction of order	Pages 150-155 & 171-172of Boyce-DiPrima, 10th Edition
9	Mar. 03	Summary of the concepts and results on 2nd order linear homogeneous ODEs. Application of the method of reduction of order to an example directly and using Abel's Theorem	-
Exam 1	Mar. 04
10	Mar. 05	Non-homogeneous 2nd order linear homogeneous ODEs: General form of the solution, variation of parameters, the Green's function	Pages 186-192 of Boyce-DiPrima, 10th Edition
11	Mar. 10	Numerical sequences, series, and their convergence; Power series and their properties, analytic functions, Taylor series; application of power series in evaluating indefinite integrals, regular and singular points of a second order linear ODE, existence of power series solutions about regular points, application of power series in solving y''-y=0.	Pages 247-258 of Boyce-DiPrima, 10th Edition
12	Mar. 12	Application of the method of power series for solving ODEs: An equation with polynomial solutions, Chebyshev's equation and Chebyshev polynomials	Pages 259-263 of Boyce-DiPrima, 10th Edition
13	Mar. 17	Integral transforms and their linearity, Laplace transform, a sufficient condition for the existence of the Laplace transform (piecewise continuous functions or exponential order), Laplace transform of polynomials, Laplace transform of the derivatives of a function, the motivation for using Laplace transform for solving differential equations	Pages 309-319 of Boyce-DiPrima, 10th Edition
14	Mar. 19	Various properties of Laplace transform, Laplace transform of the exponential functions, sine and cosine functions, and their products with polynomials, statement of the convolution theorem	Pages 314, 332, 350-352 of Boyce-DiPrima, 10th Edition
15	Mar. 24	Inverse and inverse Laplace transform and their application in solving differential equations, calculating inverse Laplace transform or rational functions using the method of partial fractions and the convolution theorem, unit step functions and their application in describing piecewise continuous functions	Pages 320-329 of Boyce-DiPrima, 10th Edition
16	Mar. 26	Laplace transform of the product of a function and a unit step function, application of method of Laplace transform in solving a force harmonic oscillator that involves a step function (truncated resonance); Systems of ODEs and their relationship with single ODEs of higher order; the importance of systems of 1st order linear ODEs	Pages 329-340, 352-354 of Boyce-DiPrima, 10th Edition
17	Mar. 31	Matrix form of the systems of 1st order linear ODEs, Statement of the existence and uniqueness theorem for solution of a system of 1st order linear ODEs, Wronskian of n solutions of a homogeneous system, Fundamental set of solutions and their existence, the general solution of a homogeneous systems of 1st order linear ODEs.	Pages 390-394 of Boyce-DiPrima, 10th Edition
18	Apr. 02	Homogeneous systems of 1st order linear ODEs with constant coefficients, the eigenvalue problem for n x n matrices and its application in solving such systems, the case of n distinct real eigenvalues	Pages 396-405 of Boyce-DiPrima, 10th Edition
Spring Break
19	Apr. 14	Homogeneous systems of n first order linear ODEs with constant coefficients: The case of n distinct complex eigenvalues	Pages 408-420 of Boyce-DiPrima, 10th Edition
20	Apr. 16	Homogeneous systems of n first order linear ODEs with constant coefficients: The case of repeated eigenvalues, Fundamental matrices, non-homogeneous systems of first order linear ODEs, and variation of parameters	Pages 429-434, 421-423, 440-444 of Boyce-DiPrima, 10th Edition
21	Apr. 21	Non-homogeneous systems of first order linear ODEs, initial-value problem, application of the method of variation of parameters, fundamental matrix for a homogeneous system with constant coefficient as the exponential of a matrix	Pages 440-447 & 421-427 of Boyce-DiPrima, 10th Edition
Exam 2
22	Apr. 28	Construction of the fundamental matrix for a homogeneous system with constant coefficient as the exponential of a matrix, exponential of a diagonalizable matrix, solution of  a homogeneous system using diagonalization method	Pages 423-427 of Boyce-DiPrima, 10th Edition
23	Apr. 30	Boundary-value problems for 2nd order linear ODEs, examples with no solutions, infinitely many solutions, and unique solutions, eigenvalue problems for the second derivative operator with vanishing Dirichlet boundary conditions; The heat equation with general boundary and initial conditions, the case of vanishing Dirichlet boundary conditions, solution by separation of variables, the motivatioon for Fourier sine series	Pages 589-595 & 623-627 of Boyce-DiPrima, 10th Edition
24	May 05	Review of the solution of the heat equation with Dirichlet boundary conditions. The heat equation with non-homogeneous boundary conditions, an example	Pages 623-635 of Boyce-DiPrima, 10th Edition
25	May 07	Heat equation with Neumann boundary conditions, and the Fourire cosine series; The real Fourier series.	Pages 635-639 of Boyce-DiPrima, 10th Edition
26	May 12	Odd, even and periodic functions, Dirichlet's theorem on Fourier series (Fourier convergence theorem), an example; the wave equation in 1+1 dimensions, the vibrating string	Pages 607-622 & 643-652of Boyce-DiPrima, 10th Edition
27	May 14	Application of the solution of wave equation for vibrating sting, 1+1 dimensional wave equation for infinite sting and Dalamberte's solution	Pages 654-655 of Boyce-DiPrima, 10th Edition

Note: The pages from the textbook listed above may not include some of the subjects covered in the lectures.