Math 204, Fall 2016

Sections 2 & 3

Topics Covered in Lectures by Ali Mostafazadeh

Lecture Number	Date	Content	Corresponding Reading material
1	Sep. 26	Introduction to and classification of differential equations: ODEs, PDEs, order of a differential equation, linear and  nonlinear equations, Definition of a solution of an ODE in an interval	Sections 1.1-1.3 of Boyce & DiPrima, 10th Edition
2	Sep. 28	Linear first order ODEs: Methods of integrating factor and variation of paramaters; derivation of the solution of the general initial-value problem for 1st order linear equations	Section 2.1 of Boyce & DiPrima, 10th Edition
3	Oct. 03	Nonlinear first order ODEs: Bernouli's Equation, Exact Equations: Definition and a characterization theorem for exact equations (exactness test), an example.	Pages 77-78 & 95-98 Boyce-DiPrima, 10th Edition
4	Oct. 05	Integrating Factors (turning a first order equation into an exact equation), an example, solving linear 1st ODEs by turning them into exact equations; An initial-value linear 1st order  ODE with no solution, Existence and uniqueness theorem for the solution of the first order linear ODEs	Pages 99-100 & 69-70 of Boyce-DiPrima, 10th Edition
5	Oct. 10	Existence and uniqueness theorem for the solution of the first order nonlinear ODEs, an initial-value problem with infinitly many solutions; 2nd order ODEs, Linear 2nd order ODEs, homogeneous and non-homogeneous equations, relevance of linear algebra, function spaces (C^n)	Pages 70-76,  112-113 & 137-138 of Boyce-DiPrima, 10th Edition
6	Oct. 12	Review of linear algebra: Vector spaces, subspace, linear combination, span, linear independence, basis, space of polynomials, linear operators (maps), 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 2nd order ODEs with constant coefficients	Chapter 3, Sections 1, 2, 4, 5 & Chapter 4, Sections 2, 3 of Serge Lang's "Introduction to Linear Algebra"
7	Oct. 17	Solving 2nd order linear homogeneous ODEs with constant coefficient by reduction to first order linear equations, the characteristic equation, solution for the cases of distinct and repeated real roots, review of complex numbers and real exponential function, the complex power series and the definition of the exponential of a complex number.	Pages 138-143 of Boyce-DiPrima, 10th Edition
8	Oct. 19	Euler's formula, superposition principle, 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), the definition of the Wronskian of two differentiable functions, Wronskian as a measure of linear-dependence, Abel's theorem and its consequences, fundamental set of solutions, the general form of the solution for a 2nd order linear homogeneous ODEs.	Pages 145-167 of Boyce-DiPrima, 10th Edition
9	Oct. 24	Application of Abel's Theorem in determining the general solution of a 2nd order linear homogeneous ODE using a given nonzero solution, method of reduction of order; Non-homogeneous 2nd order linear ODEs: General form of the solution	Pages 171-177 of Boyce-DiPrima, 10th Edition
10	Oct. 26	Solving a non-homogeneous linear ODE using the method of variation of parameters, the Green's function, an example.	Pages 186-192 of Boyce-DiPrima, 10th Edition
11	Oct. 31	Sequences, series, and their convergence; the Ratio test, Power series and their properties, analytic functions, Taylor series; regular and singular points of a second order linear ODE, existence of power series solutions about regular points	Pages 247-255 of Boyce-DiPrima, 10th Edition
12	Nov. 02	Application of the method of power series for solving 2nd order linear homogeneous ODEs	Pages 255-263 of Boyce-DiPrima, 10th Edition
	Nov. 05	Midterm Exam 1 (Coverage: Material of Lectures 1-10)
13	Nov. 07	Integral transforms and their linearity, Laplace transform, piecewise continuous functions of exponential order, existence theorem for the Laplace transform of  piecewise continuous functions of exponential order and its proof, Laplace transform of polynomials, Laplace transform of the derivative of a function, the motivation for using Laplace transform for solving differential equations	Pages 309-317 of Boyce-DiPrima, 10th Edition
14	Nov. 09	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, the inverse Laplace transform	Pages 318-322, 332, & 350-352 of Boyce-DiPrima, 10th Edition
15	Nov. 14	Application of Laplace transform in solving differential equations, unit step functions and their application in describing piecewise continuous functions, Laplace transform of the step function	Pages 327-332 of Boyce-DiPrima, 10th Edition
16	Nov. 16	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, Matrix form of a systems of 1st order linear ODEs, homogeneous systems	Pages 336-340, 359-361, & 390-394 of Boyce-DiPrima, 10th Edition
17	Nov. 21	Canonical form of a system of 1st order linear ODEs, statement of the existence and uniqueness theorem for solution of a system of 1st order linear ODEs, Superposition principle, 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 363 & 390-394 of Boyce-DiPrima, 10th Edition
18	Nov. 23	Homogeneous systems of n first order linear ODEs with constant coefficients: The general setup and solution for the case of n distinct real eigenvalues	Pages 396-405 of Boyce-DiPrima, 10th Edition
19	Nov. 28	Homogeneous systems of n first order linear ODEs with constant coefficients: The case of complex and repeated eigenvalues. Definition of a fundamental matrix	Pages 408-417 & 429-432 of Boyce-DiPrima, 10th Edition
20	Nov. 30	Properties of fundamental matrices, a fundamental matrix for systems with constant coefficient, exponential of an n x n matrix, exponential of a diagonalizable matrix, an example	Pages 421-427 Boyce-DiPrima, 10th Edition
21	Dec. 05	General solution of the non-homogeneous system of first order linear ODEs, Method of variation of parameters and the matrix Green's function, the solution of  the initial-value problem, the case of constant coefficient, an example	Pages 440-447 Boyce-DiPrima, 10th Edition
22	Dec. 07	Solution of an initial-value problem for a non-homogeneous system of linear ODEs with a constant coefficient matrix A by evaluating the fundamental matrix e^(tA), the solution of systems of linear ODEs with a constant diagonalizable coefficient matrix using the diagonalization method; The boundary-value problem for 2nd order linear ODEs, examples with no solution, infinitely many solution, and a unique solution	Pages 589-592 Boyce-DiPrima, 10th Edition
	Dec. 10	Midterm Exam 2 (Coverage: Material of Lectures 11-22)
23	Dec. 12	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 motivation for Fourier sine series	Pages 592-595 & 623-627 of Boyce-DiPrima, 10th Edition
24	Dec. 14	Review of the solution of the heat equation with Dirichlet boundary conditions, determination of the coefficients of the series solution; The heat equation with non-homogeneous boundary conditions	Pages 623-635 of Boyce-DiPrima, 10th Edition
25	Dec. 19	Heat equation with Neumann boundary conditions and Fourier cosine series, the even and odd extensions of a function defined on [0,L], the motivation for the real Fourier series of a function defined on [-L,L).	Pages 635-638 & 614-619 of Boyce-DiPrima, 10th Edition
26	Dec. 21	Solution of the heat equation with Neumann boundary conditions  for a sinusoidal initial condition, periodic extension of a function defined on a finite interval, Dirichlet's theorem on Fourier series (Fourier convergence theorem), an example.	Pages 607-622 of Boyce-DiPrima, 10th Edition
27	Dec. 26	The wave equation in 1+1 dimensions, the vibrating string	Pages  643-652 of Boyce-DiPrima, 10th Edition
28	Dec. 28	1+1 dimensional wave equation for infinite string and Dalembert's solution	Pages 654-655 of Boyce-DiPrima, 10th Edition
	Jan. 07	Final Exam (Coverage: Material of Lectures 1-28 with more emphasize given to the content of Lectures 23-28)

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