Table of Contents
Preface ............................................................................................................................................ 3
Outline ........................................................................................................................................... iv
Basic Concepts ............................................................................................................................... 1
Introduction ............................................................................................................................................... 1
Definitions ................................................................................................................................................. 2
Direction Fields ......................................................................................................................................... 8
Final Thoughts ........................................................................................................................................ 19
First Order Differential Equations ............................................................................................. 20
Introduction ............................................................................................................................................. 20
Linear Differential Equations ................................................................................................................... 21
Separable Differential Equations .............................................................................................................. 34
Exact Differential Equations .................................................................................................................... 45
Bernoulli Differential Equations .............................................................................................................. 56
Substitutions ............................................................................................................................................ 63
Intervals of Validity ................................................................................................................................ 72
Modeling with First Order Differential Equations ................................................................................... 77
Equilibrium Solutions .............................................................................................................................. 90
Euler’s Method ........................................................................................................................................ 94
Second Order Differential Equations ....................................................................................... 102
Introduction ........................................................................................................................................... 102
Basic Concepts ...................................................................................................................................... 104
Real, Distinct Roots ............................................................................................................................... 109
Complex Roots ...................................................................................................................................... 113
Repeated Roots ...................................................................................................................................... 118
Reduction of Order ................................................................................................................................ 122
Fundamental Sets of Solutions ............................................................................................................... 126
More on the Wronskian .......................................................................................................................... 131
Nonhomogeneous Differential Equations .............................................................................................. 137
Undetermined Coefficients .................................................................................................................... 139
Variation of Parameters .......................................................................................................................... 156
Mechanical Vibrations ........................................................................................................................... 162
Laplace Transforms ................................................................................................................... 181
Introduction ........................................................................................................................................... 181
The Definition ....................................................................................................................................... 183
Laplace Transforms ............................................................................................................................... 187
Inverse Laplace Transforms ................................................................................................................... 191
Step Functions ....................................................................................................................................... 202
Solving IVP’s with Laplace Transforms ................................................................................................ 215
Nonconstant Coefficient IVP’s .............................................................................................................. 222
IVP’s With Step Functions ..................................................................................................................... 226
Dirac Delta Function .............................................................................................................................. 233
Convolution Integrals ............................................................................................................................. 236
Systems of Differential Equations ............................................................................................ 241
Introduction ........................................................................................................................................... 241
Review : Systems of Equations .............................................................................................................. 243
Review : Matrices and Vectors .............................................................................................................. 249
Review : Eigenvalues and Eigenvectors ................................................................................................. 259
Systems of Differential Equations .......................................................................................................... 269
Solutions to Systems .............................................................................................................................. 273
Phase Plane............................................................................................................................................ 275
Real, Distinct Eigenvalues ..................................................................................................................... 280
Complex Eigenvalues ............................................................................................................................. 290
Repeated Eigenvalues ............................................................................................................................ 296
Nonhomogeneous Systems .................................................................................................................... 303
Laplace Transforms ............................................................................................................................... 307
Modeling ............................................................................................................................................... 309
Series Solutions to Differential Equations ............................................................................... 318
Introduction ........................................................................................................................................... 318
Review : Power Series........................................................................................................................... 319
Review : Taylor Series ........................................................................................................................... 327
Series Solutions to Differential Equations ............................................................................................. 330
Euler Equations ..................................................................................................................................... 340
Higher Order Differential Equations ....................................................................................... 346
Introduction ........................................................................................................................................... 346
Basic Concepts for nth Order Linear Equations ...................................................................................... 347
Linear Homogeneous Differential Equations ......................................................................................... 350
Undetermined Coefficients .................................................................................................................... 355
Variation of Parameters .......................................................................................................................... 357
Laplace Transforms ............................................................................................................................... 363
Systems of Differential Equations .......................................................................................................... 365
Series Solutions ..................................................................................................................................... 370
Boundary Value Problems & Fourier Series ........................................................................... 374
Introduction ........................................................................................................................................... 374
Boundary Value Problems .................................................................................................................... 375
Eigenvalues and Eigenfunctions ............................................................................................................ 381
Periodic Functions, Even/Odd Functions and Orthogonal Functions..................................................... 398
Fourier Sine Series ................................................................................................................................ 406
Fourier Cosine Series ............................................................................................................................. 417
Fourier Series ........................................................................................................................................ 426
Convergence of Fourier Series ............................................................................................................... 434
Partial Differential Equations ................................................................................................... 440
Introduction ........................................................................................................................................... 440
The Heat Equation ................................................................................................................................. 442
The Wave Equation ............................................................................................................................... 449
Terminology .......................................................................................................................................... 451
Separation of Variables .......................................................................................................................... 454
Solving the Heat Equation ...................................................................................................................... 465
Heat Equation with Non-Zero Temperature Boundaries ........................................................................ 478
Laplace’s Equation ................................................................................................................................ 481
Vibrating String ..................................................................................................................................... 492
Summary of Separation of Variables ..................................................................................................... 495
Preface
Here are my online notes for my differential equations course that I teach here at LamarUniversity. 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.
