Paul's Online Math Notes
Differential Equations (Notes)   [Notes]

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August 7, 2018

Differential Equations - Notes
  Next Chapter Basic Concepts
  Next Section Basic Concepts (Introduction)
Differential Equations (Math 3301)

Here are my online notes for my differential equations course that I teach here at Lamar University.  Despite the fact that these are my “class notes”, they should be accessible to anyone wanting to learn how to solve differential equations or needing a refresher on differential equations. 


I’ve tried to make these notes as self contained as possible and so all the information needed to read through them is either from a Calculus or Algebra class or contained in other sections of the notes.


A couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. 


  1. Because I wanted to make this a fairly complete set of notes for anyone wanting to learn differential equations I have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here.  You will need to find one of your fellow class mates to see if there is something in these notes that wasn’t covered in class.

  2. In general I try to work problems in class that are different from my notes.  However, with Differential Equation many of the problems are difficult to make up on the spur of the moment and so in this class my class work will follow these notes fairly close as far as worked problems go.  With that being said I will, on occasion, work problems off the top of my head when I can to provide more examples than just those in my notes.  Also, I often don’t have time in class to work all of the problems in the notes and so you will find that some sections contain problems that weren’t worked in class due to time restrictions.

  3. Sometimes questions in class will lead down paths that are not covered here.  I try to anticipate as many of the questions as possible in writing these up, but the reality is that I can’t anticipate all the questions.  Sometimes a very good question gets asked in class that leads to insights that I’ve not included here.  You should always talk to someone who was in class on the day you missed and compare these notes to their notes and see what the differences are.

  4. This is somewhat related to the previous three items, but is important enough to merit its own item.  THESE NOTES ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!!  Using these notes as a substitute for class is liable to get you in trouble. As already noted not everything in these notes is covered in class and often material or insights not in these notes is covered in class.


Here is a listing and brief description of the material in this set of notes.


Basic Concepts

Definitions  Some of the common definitions and concepts in a differential equations course

Direction Fields  An introduction to direction fields and what they can tell us about the solution to a differential equation.

Final Thoughts  A couple of final thoughts on what we will be looking at throughout this course.



First Order Differential Equations

Linear Equations  Identifying and solving linear first order differential equations.

Separable Equations  Identifying and solving separable first order differential equations.  We’ll also start looking at finding the interval of validity from the solution to a differential equation.

Exact Equations  Identifying and solving exact differential equations.  We’ll do a few more interval of validity problems here as well.

Bernoulli Differential Equations  In this section we’ll see how to solve the Bernoulli Differential Equation.  This section will also introduce the idea of using a substitution to help us solve differential equations.

Substitutions  We’ll pick up where the last section left off and take a look at a couple of other substitutions that can be used to solve some differential equations that we couldn’t otherwise solve.

Intervals of Validity  Here we will give an in-depth look at intervals of validity as well as an answer to the existence and uniqueness question for first order differential equations.

Modeling with First Order Differential Equations  Using first order differential equations to model physical situations.  The section will show some very real applications of first order differential equations.

Equilibrium Solutions  We will look at the behavior of equilibrium solutions and autonomous differential equations.

Euler’s Method  In this section we’ll take a brief look at a method for approximating solutions to differential equations.


            Second Order Differential Equations

Basic Concepts  Some of the basic concepts and ideas that are involved in solving second order differential equations.

Real Roots  Solving differential equations whose characteristic equation has real roots.

Complex Roots  Solving differential equations whose characteristic equation complex real roots.

Repeated Roots  Solving differential equations whose characteristic equation has repeated roots.

Reduction of Order  A brief look at the topic of reduction of order.  This will be one of the few times in this chapter that non-constant coefficient differential equation will be looked at.

Fundamental Sets of Solutions  A look at some of the theory behind the solution to second order differential equations, including looks at the Wronskian and fundamental sets of solutions.

More on the Wronskian  An application of the Wronskian and an alternate method for finding it.

Nonhomogeneous Differential Equations  A quick look into how to solve nonhomogeneous differential equations in general.

Undetermined Coefficients  The first method for solving nonhomogeneous differential equations that we’ll be looking at in this section.

Variation of Parameters  Another method for solving nonhomogeneous differential equations.

Mechanical Vibrations  An application of second order differential equations.  This section focuses on mechanical vibrations, yet a simple change of notation can move this into almost any other engineering field.


            Laplace Transforms

The Definition  The definition of the Laplace transform.  We will also compute a couple Laplace transforms using the definition.

Laplace Transforms  As the previous section will demonstrate, computing Laplace transforms directly from the definition can be a fairly painful process.  In this section we introduce the way we usually compute Laplace transforms.

Inverse Laplace Transforms  In this section we ask the opposite question.  Here’s a Laplace transform, what function did we originally have?

Step Functions  This is one of the more important functions in the use of Laplace transforms.  With the introduction of this function the reason for doing Laplace transforms starts to become apparent.

Solving IVP’s with Laplace Transforms  Here’s how we used Laplace transforms to solve IVP’s.

Nonconstant Coefficient IVP’s  We will see how Laplace transforms can be used to solve some nonconstant coefficient IVP’s

IVP’s with Step Functions  Solving IVP’s that contain step functions.  This is the section where the reason for using Laplace transforms really becomes apparent.

Dirac Delta Function  One last function that often shows up in Laplace transform problems.

Convolution Integral  A brief introduction to the convolution integral and an application for Laplace transforms.

Table of Laplace Transforms  This is a small table of Laplace Transforms that we’ll be using here.


            Systems of Differential Equations

Review : Systems of Equations  The traditional starting point for a linear algebra class.  We will use linear algebra techniques to solve a system of equations.

Review : Matrices and Vectors  A brief introduction to matrices and vectors.  We will look at arithmetic involving matrices and vectors, inverse of a matrix, determinant of a matrix, linearly independent vectors and systems of equations revisited.

Review : Eigenvalues and Eigenvectors  Finding the eigenvalues and eigenvectors of a matrix.  This topic will be key to solving systems of differential equations.

Systems of Differential Equations  Here we will look at some of the basics of systems of differential equations.

Solutions to Systems  We will take a look at what is involved in solving a system of differential equations.

Phase Plane  A brief introduction to the phase plane and phase portraits.

Real Eigenvalues  Solving systems of differential equations with real eigenvalues.

Complex Eigenvalues  Solving systems of differential equations with complex eigenvalues.

Repeated Eigenvalues  Solving systems of differential equations with repeated eigenvalues.

Nonhomogeneous Systems  Solving nonhomogeneous systems of differential equations using undetermined coefficients and variation of parameters.

Laplace Transforms  A very brief look at how Laplace transforms can be used to solve a system of differential equations.

Modeling  In this section we’ll take a quick look at some extensions of some of the modeling we did in previous sections that lead to systems of equations.


            Series Solutions

Review : Power Series  A brief review of some of the basics of power series.

Review : Taylor Series  A reminder on how to construct the Taylor series for a function.

Series Solutions  In this section we will construct a series solution for a differential equation about an ordinary point.

Euler Equations  We will look at solutions to Euler’s differential equation in this section.


            Higher Order Differential Equations

Basic Concepts for nth Order Linear Equations  We’ll start the chapter off with a quick look at some of the basic ideas behind solving higher order linear differential equations.

Linear Homogeneous Differential Equations  In this section we’ll take a look at extending the ideas behind solving 2nd order differential equations to higher order.

Undetermined Coefficients  Here we’ll look at undetermined coefficients for higher order differential equations.

Variation of Parameters  We’ll look at variation of parameters for higher order differential equations in this section.

Laplace Transforms  In this section we’re just going to work an example of using Laplace transforms to solve a differential equation on a 3rd order differential equation just so say that we looked at one with order higher than 2nd.

Systems of Differential Equations  Here we’ll take a quick look at extending the ideas we discussed when solving 2 x 2 systems of differential equations to systems of size 3 x 3.

Series Solutions  This section serves the same purpose as the Laplace Transform section.  It is just here so we can say we’ve worked an example using series solutions for a differential equations of order higher than 2nd.


Boundary Value Problems & Fourier Series

Boundary Value Problems  In this section we’ll define the boundary value problems as well as work some basic examples.
Eigenvalues and Eigenfunctions  Here we’ll take a look at the eigenvalues and eigenfunctions for boundary value problems.

Periodic Functions and Orthogonal Functions  We’ll take a look at periodic functions and orthogonal functions in section.

Fourier Sine Series  In this section we’ll start looking at Fourier Series by looking at a special case : Fourier Sine Series.

Fourier Cosine Series  We’ll continue looking at Fourier Series by taking a look at another special case : Fourier Cosine Series.

Fourier Series  Here we will look at the full Fourier series.

Convergence of Fourier Series  Here we’ll take a look at some ideas involved in the just what functions the Fourier series converge to as well as differentiation and integration of a Fourier series.


Partial Differential Equations

The Heat Equation  We do a partial derivation of the heat equation in this section as well as a discussion of possible boundary values.
The Wave Equation  Here we do a partial derivation of the wave equation.
Terminology  In this section we take a quick look at some of the terminology used in the method of separation of variables.
Separation of Variables  We take a look at the first step in the method of separation of variables in this section.  This first step is really the step that motivates the whole process.

Solving the Heat Equation  In this section we go through the complete separation of variables process and along the way solve the heat equation with three different sets of boundary conditions.

Heat Equation with Non-Zero Temperature Boundaries  Here we take a quick look at solving the heat equation in which the boundary conditions are fixed, non-zero temperature conditions.
Laplace’s Equation  We discuss solving Laplace’s equation on both a rectangle and a disk in this section.
Vibrating String  Here we solve the wave equation for a vibrating string.
Summary of Separation of Variables  In this final section we give a quick summary of the method of separation of variables.

  Next Section Basic Concepts (Introduction)
  Next Chapter Basic Concepts

Differential Equations (Notes)    [Notes]

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