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

Hw 2 Due Mon Feb 10 Solution

Hw 3 Due Mon Feb 17 Solution

Hw 4 Due Mon Feb 24 Solution

 

Midterm1 Solution

 

Hw5 Due Wed March 5 Solution

Hw6 Due Wed March 12 Solution

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

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):

  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