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:


Hw 1 Due Mon Feb 3 Solution Matlab File

Hw 2 Due Mon Feb 10 Solution Matlab File

Hw 3 Due Mon Feb 17 Solution Figure MatlabFile

Quiz 1 Solution

Hw 4 Due Mon Feb 24 Solution MatLabFile


Midterm1 Solution


Hw5 Due Wed March 5 Solution

Hw6 Due Wed March 12 Solution MatlabFile

Hw7 Due Wed March 26 Solution

Hw8 Due Wed April 2nd Solution


Practice Problems for Midterm 2


Midterm2 Solution


Hw9 Due Friday April 11 Solution

Hw10 Due Friday April 18 Solution

Hw11 Due Friday April 25 Solution

Hw12 Due Friday May 2nd Solution MatlabFile

Hw13 Due Friday May 9 Solution MatlabFile


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):

  1. Syllabus – discussion

  2. Chapter 1: Introduction

  3. Definition of Ordinary Differential Equations and examples

  4. Applications and mathematical models


*Lecture 2 (01/24)

  1. How to construct mathematical models: Identify dependent and independent variables, choosing units, and articulate principles

  2. Example: Object falling in the atmosphere

  3. Qualitative behavior of solutions

  4. Slope/direction field: You can use this Matlab code to draw slope fields


*Lecture 3 (01/27)

  1. Section 1.2 Solutions of some differential equations

  2. Example: Population growth

  3. 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)

  1. Chapter 2: First order differential equations

  2. Section 2.1 Linear equations; Method of integrating factors

  3. Examples


*Lecture 5 (01/31):

  1. Section 2.2 Separable equations Matlab File

  2. Implicit solution, transforming a separable ODE into an algebraic equations


*Lecture 6 (02/03)

  1. Solution curves and the vertical line test

  2. Existence and uniqueness

  3. Transforming some ODEs into separable equations


*Lecture 7 (02/05)

  1. Section 2.3 Modeling with first order equations

  2. Example: Tank containing salt dissolved in water


*Lecture 8 (02/07)

  1. Example: A body of constant mass projected away from the Earth

  2. Section 2.4: Differences between linear and non-linear equations

  3. Existence and uniqueness, theorem


*Lecture 9 (02/10)

  1. Section 2.5: Autonomous equations and population dynamics

  2. Concavity


*Lecture 10 (02/12)

  1. Section 2.6 Exact Equations and Integrating Factors

  2. How to determine if an equation is in exact form

  3. How to find implicit solutions for exact equations


*Lecture 11 (02/14)

  1. Use of integrating factors to transform some equations into exact equations

  2. Section 2.7: Numerical Approximations: Euler's Method Matlab Files

  3. Time step

  4. Accuracy


*Lecture 12 (02/17)

  1. Accuracy of Euler's method

  2. Examples

  3. Chapter 3: Second Order Linear Equations

  4. Section 3.1: Homogeneous Equations with Constant Coefficients

  5. Linear combinations of solutions


*Lecture 13 (02/19)

  1. Exponential solutions

  2. Two parameter family of solutions

  3. Section 3.2: Solutions of Linear Homogeneous Equations; the Wronskian


*Lecture 14 (02/21)

  1. Existence and uniqueness of 2nd order linear equations

  2. Fundamental solutions

  3. The Wronskian


*Lecture 15 (02/24)

  1. Review of linear algebra

  2. General solutions of 2nd Order Linear Homogeneous Equations


*Lecture 16 (02/26)

  1. Abel's theorem and the Wronskian

  2. Section 3.3: Complex roots and the characteristic equation


*Lecture 17 (02/28)

  1. Section 3.4: Repeated roots: D'Alembert's approach

  2. Examples


*Lecture 18 (03/03)

  1. Reduction of order

  2. Examples


*Lecture 19 (03/05)

  1. Section 3.5: Non-homogeneous equations; Method of Undetermined Coefficients

  2. Non-homogeneous equations with constant coefficients and an exponential function on the right hand side


*Lecture 20 (03/07)

  1. Non-homogeneous equations with constant coefficients and an trigonometric function on the right hand side

  2. Examples


*Lecture 21 (03/10)

  1. Section 3.6: Variation of Parameters

  2. General solution to the non-homogeneous problem

  3. Examples


*Lecture 22 (03/12)

  1. Chapter 4: Higher Order Linear Equations

  2. Section 4.1 General Theory of nth Order Linear Equations

  3. Determinants of square matrices

  4. Linear Independence and the Wronskian W[y_1,y_2,...,y_n]


*Lecture 23 (03/12) By M. Busidic (Thank you!)

  1. Chapter 5: Series Solutions of Second Order Linear Equations

  2. Examples


Spring (or winter?) Break !!!


*Lecture 24 (03/24)

  1. Section 5.1 More on review of power series

  2. Radius of convergence

  3. Ratio test


*Lecture 25 (03/26)

  1. Section 5.2: Series Solutions Near an ordinary Point Part I

  2. Ordinary and singular points

  3. Power series centered at x_0

  4. Recurrence relation

  5. Examples


*Lecture 26 (03/28)

  1. Examples of Series Solutions Near Ordinary Points

  2. Graphs and plots


*Lecture 27 (03/31)

  1. Section 5.3: Series Solutions Near Ordinary Points Part II

  2. Analytical coefficient functions

  3. Example: The Legendre equation


*Lecture 28 (04/02)

  1. More examples

  2. Section 5.4: Euler Equations; Regular Singular Points

  3. Exponents and the indicial equation


*Lecture 29 (04/04)

  1. The indicial equation: Real distinct roots

  2. Double roots

  3. Complex roots


*Lecture 30 (04/07)

  1. Definition of Regular Singular Points

  2. Examples

  3. Section 5.5: Series Solutions Near a Regular Singular Point, Part I

  4. Frobenius' Method

  5. Exponent of the singularity and the recurrence relation


*Lecture 31 (04/09)

  1. Examples of Series Solutions Near Regular Singular Points

  2. Section 5.6: The general case


*Lecture 32 (04/11)

  1. The recurrence relation

  2. The indicial equation

  3. Examples


*Lecture 33 (04/14)

  1. Chapter 6: The Laplace Transform

  2. Section 6.1: Definition of the Laplace transform

  3. Improper integrals

  4. Piecewise defined functions

  5. Integral transforms, the kernel


*Lecture 34 (04/16)

  1. The Laplace transform

  2. Examples

  3. Section 6.2 : Solutions of initial value problems

  4. Properties of the Laplace transform


*Lecture 35 (04/18)

  1. Table of Elementary Laplace Transforms

  2. Computing the inverse transforms of several functions using partial fractions

  3. Solving ODEs using the Laplace transform


*Lecture 36 (04/21)

  1. Section 6.4: Differential Equations with Discontinuous Forcing Functions

  2. Applications: Simple electric circuits with a unit voltage pulse for a period of time

  3. Step functions and their Laplace transforms


*Lecture 37 (04/23)

  1. Thermal energy density

  2. Conservation of heat energy

  3. Heat flux

  4. The heat equation

  5. Fourier's law of heat conduction


*Lecture 38 (04/25)

  1. Different boundary conditions

  2. Prescribed temperatures

  3. Insulated boundaries

  4. Newton's law of cooling

  5. Method of separation of variables


*Lecture 39 (04/28)

  1. Principle of superposition

  2. Fourier sine series


*Lecture 40 (04/30)

  1. Finite Difference Numerical Approximations

  2. Polynomial approximations


*Lecture 41 (05/02)

  1. Error analysis

  2. Numerical PDEs


*Lecture 42 (05/05)

  1. Discretization of space and time


*Lecture 43 (05/07)

  1. Fourier-von Neumann stability analysis


*Lecture 44 (05/09)

  1. Numerical results of 1D and 2D heat equation