Math 204, Fall 2021
Section 4
Topics Covered in Lectures by Ali Mostafazadeh
|
Lecture Number |
Date |
Content |
Corresponding Reading material* |
|
1 |
Sep. 27 |
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 textbook (Boyce & DiPrima) |
|
2 |
Sep. 29 |
Linear first-order ODEs: Methods of integrating factor and
variation of parameters, examples, Bernoulli’s Equation. |
Section
2.1 of textbook |
|
3 |
Oct. 04 |
Solution of initial-value
problem and existence and uniqueness theorem for a linear first-order ODE;
Nonlinear first-order equations: Separable and exact equations |
Sections 2.2,
2.4, 2.6 |
|
4 |
Oct. 06 |
Characterization theorem for exact equations (exactness
test), integrating factors (turning a first-order equation into an exact
equation) |
Section
2.6 |
|
5 |
Oct. 11 |
Application of integrating
factor for solving a nonlinear first-order ODE, Existence and uniqueness
theorem for first order ODEs (without proof) |
Sections
2.6 & 2.8 |
|
6 |
Oct. 13 |
Euler’s numerical solution
of first-order ODEs, Picard’s iterative scheme for solving first-order ODEs,
Second-order ODEs: General strategy for solving second-order linear ODEs;
Review of basic concepts of linear algebra: Real vector spaces, subspaces of
a vector space, vector spaces C(U) of real-valued functions defined on an
interval U of real numbers, Cn(U) with n=0,1,2,3,…, vector space
of real-valued smooth and analytic functions defined on U |
Sections
2.7, 2.8 & 3.1 |
|
7 |
Oct. 18 |
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, homogeneous linear 2nd order ODEs with constant coefficients:
Reduction to two 1st order equations |
Section
3.1 |
|
8 |
Oct. 20 |
Solving homogeneous linear 2nd order ODEs with constant
coefficients by their reduction to 1st order ODEs, 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
(Euler's formula). |
Sections
3.1 & 3.3 |
|
9 |
Oct. 25 |
Solution of 2nd
order linear homogeneous ODEs with constant coefficients when the roots of
the characteristic polynomial are complex. General theory of 2nd
order linear homogeneous ODEs: Superposition principle, 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. |
Sections
3.2 & 3.3 |
|
10 |
Oct. 27 |
General solution of 2nd
order linear homogeneous ODEs with constant coefficients, determining
the general solution of a 2nd order linear homogeneous ODE using a given
nonzero solution (reduction of order). |
Sections
3.2 & 3.4 |
|
11 |
Nov. 01 |
Application of Abel’s
theorem in constructing the general solution of a 2nd
order linear homogeneous ODE using a given nonzero solution; Non-homogeneous
2nd order linear ODE’s: The structure of the general solution, determination
of the general solution using a pair of fundamental solutions of the
corresponding homogeneous equation (variation of parameters); expression for
a particular solution in terms of the Green’s function. |
Sections
3.4, 3.5 & 3.6 |
|
12 |
Nov. 03 |
Review of the method of variation of parameters and Green’s
function, initial-value problem and the
general solution of a non-homogeneous 2nd order linear ODE’s |
Sections
3.5 & 3.6 |
|
Midterm Exam 1 |
Nov. 06 |
|
|
|
13 |
Nov. 08 |
Sequences, series, and their convergence; the Ratio test,
power series and their properties, analytic functions, Taylor series |
Section 5.1 |
|
14 |
Nov. 10 |
Regular and singular points of a second order linear ODE,
existence of power series solutions about regular points, application of the
method of power series for solving 2nd order linear homogeneous ODEs |
Sections 5.2 & 5.3 |
|
Winter Break |
|
|
|
|
15 |
Nov.22 |
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 |
Sections 6.1 & 6.2 |
|
16 |
Nov. 24 |
Various properties of Laplace transform, Laplace transform
of the exponential functions, sine and cosine functions, and their products
with polynomials, unit step functions and their application in describing
piecewise continuous functions, Laplace transform of the step function, statement of the convolution theorem, and its application
in solving a 2nd-order
linear ODE (forced oscillator) |
Sections 6.3, 6.4 & 6.6 |
|
17 |
Nov. 29 |
Use of partial fractions in
evaluating inverse Laplace transforms, solution of general 2nd-order
linear ODEs with constant coefficients using the method of Laplace transform;
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, canonical form of a system of 1st
order linear ODEs, Superposition principle, Wronskian
of n solutions of a homogeneous system of n linear ODEs, Fundamental set of
solutions, Abel’s theorem for systems of ODEs. |
Sections 6.6, 7.1 & 7.4 |
|
18 |
Dec. 01 |
Canonical form of a system of 1st order linear ODEs,
statement of the existence and uniqueness theorem for solution of a
homogeneous system of 1st order linear ODEs, existence of fundamental sets of
solutions of homogeneous systems of 1st order linear ODEs, the general
solution of such a system of ODEs; Homogeneous systems of n first order
linear ODEs with constant coefficients: The relation to eigenvalue problem
for real square matrices, diagonalizable and non-diagonalizable matrices |
Sections 7.4 & 7.5 |
|
19 |
|Dec. 06 |
Homogeneous systems of n first order linear ODEs with constant
coefficients: The case where the matrix of coefficients has n distinct real
or complex eigenvalues |
Sections 7.5 & 7.6 |
|
20 |
Dec. 08 |
Homogeneous systems of n first order linear ODEs with constant
coefficients: The case where the matrix of coefficients has repeated
eigenvalues, solutions obtained using generalized eigenvectors |
Section 7.8 |
|
21 |
Dec. 13 |
Fundamental matrix for a homogeneous system of first order
linear ODEs, solution of the non-homogeneous systems of first order linear
ODEs using the method of variation of parameters, the matrix Green's
function, the solution of the initial-value problem for a
non-homogeneous system, the definition of the exponential of a square matrix,
etA as a fundamental matrix for a
homogeneous system with a constant matrix of coefficients A |
Sections 7.7 & 7.9 |
|
22 |
Dec. 15 |
Review of the solution of the
initial-value problem for a non-homogeneous system of first order linear
ODEs, calculation of etA for a diagonalizable matrix A, an example of the application of
the method of variation of parameters and the fundamental matrix etA |
Section 7.9 |
|
Midterm Exam 2 |
Dec. 19 |
|
|
|
23 |
Dec. 20 |
The boundary-value problem for 2nd order linear ODEs,
examples with no solution, infinitely many solution, and a unique solution,
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 |
Sections 10.1 & 10.5 |
|
24 |
Dec. 22 |
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. |
Sections 10.5 & 10.6 |
|
25 |
Dec. 27 |
Heat equation with Neumann boundary conditions and Fourier
cosine series, the even and odd extensions of a function defined on [0,L],
periodic extension of a function defined on a finite interval, Dirichlet's theorem on Fourier series (Fourier
convergence theorem) |
Sections 10.2-10.4 & 10.6 |
|
26 |
Dec. 29 |
Proof of the uniqueness
theorem for the solution of the heat equation with general Dirichlet boundary conditions, Heat conduction problem on
a circle (periodic boundary conditions.) |
- |
|
27 |
Jan. 03 |
The wave equation in 1+1 dimensions, the vibrating string
with a finite length |
Section 10.7 |
|
28 |
Jan. 05 |
D’Alembert solution of the wave equation in the real line
(infinite vibrating string) |
Section 1.7, Problems 15-17 |
Note:
*The
pages from the textbook listed above may not include some of the material
covered in the lectures.