Math 319 Section 003: Techniques in Ordinary Differential Equations
Schedule: MWF 9:55– 10:45 AM. Van Vleck B 130
Office hours: M 11:00 am – noon, WF 9:00 – 9:45 AM
Syllabus: Can be found here
Textbook: W. E. Boyce, R. C Diprima , Elementary Differential Equations and Boundary Value Problems, ninth ed. WILEY.
TA: Ashutosh Kumar Ashutosh's office hours: Tu, Thu 1:00 – 2:15 pm and by appointment
Homework assignments:
Hw8 Due Wed April 2nd Solution
Practice Problems for Midterm 2
Hw9 Due Friday April 11 Solution
Hw10 Due Friday April 18 Solution
Hw11 Due Friday April 25 Solution
Hw12 Due Friday May 2nd Solution
Hw13 Due Friday May 9 Solution
Practice Problems for the Final Exam
Important information:
Midterm 1: February 26th, 5:30-7:00 PM. Room: Van Vleck B130
Midterm 2: April 4th 5:30-7:00 PM. Room: Van Vleck B130
Final exam: May 16th 7:45 am - 9:45 am. Room: TBA
*See the syllabus for more information on exam policies.
Grading:
Midterm 1: 25 %
Midterm 2: 25 %
Final exam: 30 %
Homework: 10%
Quizzes: 10%
Note: Any student with a documented disability should contact me as soon as possible so that we can discuss arrangements to fit your needs.
Brief lecture outline:
* Lecture 1 (01/22):
-
Syllabus – discussion
-
Chapter 1: Introduction
-
Definition of Ordinary Differential Equations and examples
-
Applications and mathematical models
*Lecture 2 (01/24)
-
How to construct mathematical models: Identify dependent and independent variables, choosing units, and articulate principles
-
Example: Object falling in the atmosphere
-
Qualitative behavior of solutions
-
Slope/direction field: You can use this Matlab code to draw slope fields
*Lecture 3 (01/27)
-
Section 1.2 Solutions of some differential equations
-
Example: Population growth
-
Section 1.3: Classification of ODEs: Ordinary vs Partial, Systems, Order of an ODE, Explicit vs Implicit, Linear vs Non-linear
*Lecture 4 (01/29)
-
Chapter 2: First order differential equations
-
Section 2.1 Linear equations; Method of integrating factors
-
Examples
*Lecture 5 (01/31):
-
Section 2.2 Separable equations Matlab File
-
Implicit solution, transforming a separable ODE into an algebraic equations
*Lecture 6 (02/03)
-
Solution curves and the vertical line test
-
Existence and uniqueness
-
Transforming some ODEs into separable equations
*Lecture 7 (02/05)
-
Section 2.3 Modeling with first order equations
-
Example: Tank containing salt dissolved in water
*Lecture 8 (02/07)
-
Example: A body of constant mass projected away from the Earth
-
Section 2.4: Differences between linear and non-linear equations
-
Existence and uniqueness, theorem
*Lecture 9 (02/10)
-
Section 2.5: Autonomous equations and population dynamics
-
Concavity
*Lecture 10 (02/12)
-
Section 2.6 Exact Equations and Integrating Factors
-
How to determine if an equation is in exact form
-
How to find implicit solutions for exact equations
*Lecture 11 (02/14)
-
Use of integrating factors to transform some equations into exact equations
-
Section 2.7: Numerical Approximations: Euler's Method Matlab Files
-
Time step
-
Accuracy
*Lecture 12 (02/17)
-
Accuracy of Euler's method
-
Examples
-
Chapter 3: Second Order Linear Equations
-
Section 3.1: Homogeneous Equations with Constant Coefficients
-
Linear combinations of solutions
*Lecture 13 (02/19)
-
Exponential solutions
-
Two parameter family of solutions
-
Section 3.2: Solutions of Linear Homogeneous Equations; the Wronskian
*Lecture 14 (02/21)
-
Existence and uniqueness of 2nd order linear equations
-
Fundamental solutions
-
The Wronskian
*Lecture 15 (02/24)
-
Review of linear algebra
-
General solutions of 2nd Order Linear Homogeneous Equations
*Lecture 16 (02/26)
-
Abel's theorem and the Wronskian
-
Section 3.3: Complex roots and the characteristic equation
*Lecture 17 (02/28)
-
Section 3.4: Repeated roots: D'Alembert's approach
-
Examples
*Lecture 18 (03/03)
-
Reduction of order
-
Examples
*Lecture 19 (03/05)
-
Section 3.5: Non-homogeneous equations; Method of Undetermined Coefficients
-
Non-homogeneous equations with constant coefficients and an exponential function on the right hand side
*Lecture 20 (03/07)
-
Non-homogeneous equations with constant coefficients and an trigonometric function on the right hand side
-
Examples
*Lecture 21 (03/10)
-
Section 3.6: Variation of Parameters
-
General solution to the non-homogeneous problem
-
Examples
*Lecture 22 (03/12)
-
Chapter 4: Higher Order Linear Equations
-
Section 4.1 General Theory of nth Order Linear Equations
-
Determinants of square matrices
-
Linear Independence and the Wronskian W[y_1,y_2,...,y_n]
*Lecture 23 (03/12) By M. Busidic (Thank you!)
-
Chapter 5: Series Solutions of Second Order Linear Equations
-
Examples
Spring (or winter?) Break !!!
*Lecture 24 (03/24)
-
Section 5.1 More on review of power series
-
Radius of convergence
-
Ratio test
*Lecture 25 (03/26)
-
Section 5.2: Series Solutions Near an ordinary Point Part I
-
Ordinary and singular points
-
Power series centered at x_0
-
Recurrence relation
-
Examples
*Lecture 26 (03/28)
-
Examples of Series Solutions Near Ordinary Points
-
Graphs and plots
*Lecture 27 (03/31)
-
Section 5.3: Series Solutions Near Ordinary Points Part II
-
Analytical coefficient functions
-
Example: The Legendre equation
*Lecture 28 (04/02)
-
More examples
-
Section 5.4: Euler Equations; Regular Singular Points
-
Exponents and the indicial equation
*Lecture 29 (04/04)
-
The indicial equation: Real distinct roots
-
Double roots
-
Complex roots
*Lecture 30 (04/07)
-
Definition of Regular Singular Points
-
Examples
-
Section 5.5: Series Solutions Near a Regular Singular Point, Part I
-
Frobenius' Method
-
Exponent of the singularity and the recurrence relation
*Lecture 31 (04/09)
-
Examples of Series Solutions Near Regular Singular Points
-
Section 5.6: The general case
*Lecture 32 (04/11)
-
The recurrence relation
-
The indicial equation
-
Examples
*Lecture 33 (04/14)
-
Chapter 6: The Laplace Transform
-
Section 6.1: Definition of the Laplace transform
-
Improper integrals
-
Piecewise defined functions
-
Integral transforms, the kernel
*Lecture 34 (04/16)
-
The Laplace transform
-
Examples
-
Section 6.2 : Solutions of initial value problems
-
Properties of the Laplace transform
*Lecture 35 (04/18)
-
Table of Elementary Laplace Transforms
-
Computing the inverse transforms of several functions using partial fractions
-
Solving ODEs using the Laplace transform
*Lecture 36 (04/21)
-
Section 6.4: Differential Equations with Discontinuous Forcing Functions
-
Applications: Simple electric circuits with a unit voltage pulse for a period of time
-
Step functions and their Laplace transforms
*Lecture 37 (04/23)
-
Thermal energy density
-
Conservation of heat energy
-
Heat flux
-
The heat equation
-
Fourier's law of heat conduction
*Lecture 38 (04/25)
-
Different boundary conditions
-
Prescribed temperatures
-
Insulated boundaries
-
Newton's law of cooling
-
Method of separation of variables
*Lecture 39 (04/28)
-
Principle of superposition
-
Fourier sine series
*Lecture 40 (04/30)
-
Finite Difference Numerical Approximations
-
Polynomial approximations
*Lecture 41 (05/02)
-
Error analysis
-
Numerical PDEs
*Lecture 42 (05/05)
-
Discretization of space and time
*Lecture 43 (05/07)
-
Fourier-von Neumann stability analysis
*Lecture 44 (05/09)
-
Numerical results of 1D and 2D heat equation