Math 204, Fall 2014

Topics Covered in Section 3 (taught by Ali Mostafazadeh)

 

 

Lecture Number

Date

Content

Corresponding Reading material

1

Sep. 15

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, Linear First order ODEs: Introduction to the method of integrating factor Section 1.1-1.3 & pages 36-37 of Section 2.1 of Boyce & DiPrima, 10th Edition;

2

Sep. 17

 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; examples, Bernouli's Equation Section 2.1 of Boyce & DiPrima, 10th Edition

3

Sep. 22

Exact Equations: Definition and a characterization theorem for exact equations (exactness test), an example Pages 95-99 Boyce-DiPrima, 10th Edition

4

Sep. 24

Integrating Factors (turning a first order equation into an exact equation), an example, solving linear 1st ODEs by turning them into exam 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

Sep. 29

Existence and uniqueness theorem for the solution of the first order nonlinear ODEs; 2nd order ODEs, Linear 2nd order ODEs, homogeneous and non-homogeneous equations; Review of linear algebra: Vector spaces, subspace, linear combination, span, linear independence, 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. Pages 70-76 & 137-138 of Boyce-DiPrima, 10th Edition

6

Oct. 01

 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, review of complex numbers and exponential function of a real variable Pages 138-140 of Boyce-DiPrima, 10th Edition

Kurban Bayramę

Oct.

03-07

    

7

Oct. 08

Summary of the results on the solution of 2nd order linear homogeneous ODEs with constant coefficient, algebra of 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),  superposition principle, the motivation for the concept of the Wronskian. Pages 139-143, 167-170, 158-164 & 145-148 of Boyce-DiPrima, 10th Edition

8

Oct. 13

The definition of the Wronskian, Wronskian as a measure of linear-dependence of two differentiable functions, Abel's theorem and its consequences, Fundamental set of solutions, the general form of the solution for a 2nd order linear homogeneous ODE with continuous coefficients; the method of reduction of order Pages 149-155 & 171-172of Boyce-DiPrima, 10th Edition

9

Oct. 15

Application of the method of reduction of order to an example, 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 

10

Oct. 20 

Using Wronskian to determine the general solution of a 2nd order linear homogeneous ODE using a given nonzero solution. An example of the application of this method as well as the Green's function scheme for solving non-homogeneous ODEs. Basic idea for power series solution of ODEs. Pages 186-192 of Boyce-DiPrima, 10th Edition 

11

Oct. 22

 Series, and their convergence; 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, application of power series in solving y''-y=0. Pages 247-258 of Boyce-DiPrima, 10th Edition 

12

Oct. 27

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 
 

Nov. 03

Midterm Exam 1 (Coverage: Content of Lectures 01-10)  

13

Nov. 03

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

Nov. 05

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

Nov. 10

Laplace 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

Nov. 12

 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, Matrix form of a systems of 1st order linear ODEs, homogeneous systems Pages 329-340, 352-354, 359-363, 390-394 of Boyce-DiPrima, 10th Edition 

17

Nov. 17

  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; 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 Pages  378-388, 396-403 of Boyce-DiPrima, 10th Edition 

18

Nov. 19

Homogeneous systems of n first order linear ODEs with constant coefficients: The case of n distinct real or complex eigenvalues Pages 403-420 of Boyce-DiPrima, 10th Edition 

19

Nov. 24

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, the method of variation of parameters Pages 429-434, 421-423, 440-444 of Boyce-DiPrima, 10th Edition 

20

Nov. 26

Method of variation of parameters and the matrix Green's function, Fundamental matrix for a homogeneous system with constant coefficient as the exponential of a matrix, computation of the exponential of a diagonalizable matrix Pages 440-447 & 421-427 of Boyce-DiPrima, 10th Edition 

21

Dec. 01

An example of the application of the method of variation of parameters for the solution of a non-homogeneous systems of first order linear ODEs with initial conditions; 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. Pages 440-447 & 589-595 of Boyce-DiPrima, 10th Edition 

22

Dec. 03

 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 623-627 of Boyce-DiPrima, 10th Edition 

 

Dec. 08

Midterm Exam 2 (Coverage: Content of Lectures 11-20)  

23

Dec. 10

Review of the solution of the heat equation with Dirichlet boundary conditions; The heat equation with non-homogeneous boundary conditions; Heat equation with Neumann boundary conditions and Fourier cosine series Pages 623-637 of Boyce-DiPrima, 10th Edition 

24

Dec. 15

The coefficients of the Fourier cosine series, solution of the heat equation with Neumann boundary conditions  for a sinusoidal initial condition. Pages 635-639 of Boyce-DiPrima, 10th Edition 

25

Dec. 17

Odd, even and periodic functions, the real Fourier series, Dirichlet's theorem on Fourier series (Fourier convergence theorem), an example Pages 607-622 of Boyce-DiPrima, 10th Edition

26

Dec. 22

 The wave equation in 1+1 dimensions, the vibrating string Pages  643-652 of Boyce-DiPrima, 10th Edition 

27

Dec. 24

1+1 dimensional wave equation for infinite sting and Dalamberte's solution Pages 654-655 of Boyce-DiPrima, 10th Edition 

 

Jan. 03

Final Exam  (Coverage: Content of Lectures 1-27)  

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