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DIFFERENTIAL
EQUATIONS
First Order Differential Equations
Paul Dawkins
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View Full Document Differential Equations
© 2007 Paul Dawkins
i
http://tutorial.math.lamar.edu/terms.aspx
Table of Contents
Preface .
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2
First Order Differential Equations .
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3
Introduction .
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3
Linear Differential Equations .
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4
Separable Differential Equations .
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17
Exact Differential Equations .
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28
Bernoulli Differential Equations .
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39
Substitutions .
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46
Intervals of Validity .
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55
Modeling with First Order Differential Equations .
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60
Equilibrium Solutions .
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73
Euler’s Method .
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77
Differential Equations
© 2007 Paul Dawkins
ii
http://tutorial.math.lamar.edu/terms.aspx
Preface
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!!
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This note was uploaded on 11/13/2011 for the course MATH 72 taught by Professor Quach during the Spring '09 term at San Jose City College.
 Spring '09
 quach
 Differential Equations, Calculus, Equations

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