An Introduction to Differential Equations

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An Introduction to Differential Equations


Colin Carroll

August 24, 2010
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Syllabus

Basic Info

Syllabus- Are You In the Right Room?

MATH 211 - ORDINARY DIFFERENTIAL

EQUATIONS AND LINEAR ALGEBRA

FALL 2010

TR 9:25 - 10:40am, HBH427.
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Syllabus

Basic Info

Syllabus- How To Get In Touch

Instructor: Colin Carroll

Contact Info: Office: HB 447, Phone: x4598, E-mail:[email protected]

Office Hours: Monday, Wednesday and Friday, 4-5pm and byappointment.

Course Webpage: http://math.rice.edu/ cc11
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Syllabus

Basic Info






A I d i Diff i l E i


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A I t d ti t Diff ti l E ti
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Basic Info








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Textbooks

Syllabus- Textbooks

Textbook : John Polking, Albert Boggess, David ArnoldDifferential Equations, Prentice Hall, 2nd Ed.

Supplementary References:

George Simmons, Stephen KrantzDifferential Equations, McGraw Hill, WalterRudin Student Series in Advanced

Mathematics.Morris Tenenbaum and Harry PollardOrdinary Differential Equations, Dover.

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Textbooks

Syllabus- Textbooks





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Textbooks

Syllabus- Textbooks





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Textbooks

Syllabus- Textbooks





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q

Syllabus

Grading

Syllabus- Homework

Doing many problems is best way to learn ODEs.Assigned and collected once a week.

No late homework.

Lowest homework grade is dropped.

WORK TOGETHER!

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q

Syllabus

Grading

Syllabus- Homework


No late homework.


WORK TOGETHER!



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Syllabus

Grading

Syllabus- Homework


No late homework.


WORK TOGETHER!



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Syllabus

Grading

Syllabus- Homework


No late homework.


WORK TOGETHER!

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Syllabus

Grading

Syllabus- Homework

Doing many problems is best way to learn ODEs.

Assigned and collected once a week.

No late homework.


WORK TOGETHER!



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Syllabus

Grading

Syllabus- Exams

There will be two midterm exams, and a final exam.

Exams from the summer are available on my website.

http://goforward/http://find/http://goback/


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Syllabus

Grading

Syllabus- Exams

There will be two midterm exams, and a final exam.

Exams from the summer are available on my website.

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Syllabus

Grading

Grades.

Grades will be based on homeworks and exams, and worth

approximately:Homeworks: 15 %

Midterm Exam I: 20 %

Midterm Exam II: 25 %

Final Exam: 40 %


S ll b


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Syllabus

Disability Support

Syllabus- Disability Support

It is the policy of Rice University that any student with a

disability receive fair and equal treatment in this course. If youhave a documented disability that requires academicadjustments or accommodation, please speak with me duringthe first week of class. All discussions will remain confidential.Students with disabilities will also need to contact DisabilitySupport Services in the Ley Student Center.


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Syllabus

Important Dates

Syllabus- Important Dates

Tuesday, August 24: First class.

September 30-October 5: Midterm exam ITuesday, October 12: Midterm Recess- no class!

November 4-9: Midterm exam II

Thursday, November 25: Thanksgiving Recess: - no class!

Thursday, December 2: Last day of class.December 8-15: Final Exam dates.


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Syllabus

A Note on Technology


None of the work in the class will require a computer, orhopefully even a calculator. However, I plan on holding(approximately) two intro to matlab sessions during thesemester. These will be helpful in checking work and likely ifyou take any further science/engineering courses.


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Syllabus


Pause for questions, applause.


Differential Equations
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Introduction

What is an Ordinary Differential Equation?

An ordinary differential equation (also called an ODE, orDiffEQ, pronounced diffy-Q by the cool kids) is an

equation that can be written in the form

f

x,yp xq ,yI p xq ,yP p xq , . . . ,yp n q p xq

0.

In this class, you will be asked to solve a differential

equation, by which we mean find a function yp xq thatsatisfies the above equation.

This is unhelpful. Examples will help.


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Introduction




f


0.





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q

Introduction




f


0.





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q

Examples

Example

Solve the ODEyI 3x2.

From calculus we can calculate

yI dx

3x2 dx

y

x3

C.

It doesnt get any better than this.


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Examples

Example



yI dx

3x2 dx

y

x3

C.



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Examples

Example



yI dx

3x2 dx

y

x3

C.



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Examples

Harder examples.

What about solving the ODE yI y? We cannot justintegrate this, but there is a quick way to solve this.

Similarly the differential equation yP y 0 looks fairlysimple, but it will take most of the semester before we

can solve it. Well be happy just verifying the solution fornow.


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Examples

Harder examples.





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Examples

Harder examples.


y

Ae

x

.




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Examples

Harder examples.


y

Ae

x.



y A cos x Bsin x.

An Introduction to Differential EquationsDifferential Equations

S l i
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Solutions

Verifying Solutions

We wish to show that y A cos x Bsin x solvesyP y 0.

Certainly yI

A sin x Bcos x.So yP A cos x Bsin x.

Then

yP

y p A cos x Bsin xq p A cos x Bsin xq 0,

as desired.


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Solutions

Verifying Solutions


Certainly yI


Then

yP


as desired.


Solutions
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Solutions

Verifying Solutions


Certainly yI


Then

yP


as desired.


Solutions
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Solutions

Verifying Solutions


Certainly yI


Then

yP


as desired.


Solutions


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Solutions

The Nature of Solutions

Our intuition from calculus tells us that whatever wemean by general solution, it will not be unique, becauseof constants of integration.

Indeed, by general solution, we mean writing downevery solution to a differential equation- for an equationof order n, this will typically mean n constants ofintegration.

We are also often concerned about a particular solutionto an ODE. In this case, we will write down a differentialequation as well as initial conditions.


Solutions
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Solutions






Solutions
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Solutions
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An example

We will investigate the ODE

xI xsin t 2te cos t, with initial conditions xp 0 q 1.

It turns out that a general solution to the ODE is

xp tq p t2 Cq e cos t.

Plugging in the initial condition gives us the particularsolution

xp tq p t2 eq e cos t.


Solutions
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An example





Plugging in the initial condition gives us the particularsolutionxp tq p t2 eq e cos t.


Solutions
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An example





Plugging in the initial condition gives us the particularsolutionxp tq p t2 eq e cos t.


Notation
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Some Notes on Notation

Notation in differential equations can quickly become amess. I try to follow fairly standard practices.

The standard practices are sometimes confusing, but Iwould encourage you to emulate the notation used. If youstill wish to use your own on a graded assignment pleasemake your notation clear!

Some general rules: we will usually use x or t as theindependent variable, and y as the dependent variable.Unfortunately, the second choice for dependent variable isoften x.


Notation
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Notation
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Notation
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Notation, continued

As above, we will usually suppress the dependence of onevariable on another.

That is to say, rather than write

yI p xq x

c

x

yp xq,

we will writeyI x

c

x

y.


Notation
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Notation, continued



yI p xq x

c

x

yp xq,

we will writeyI x

c

x

y.


Notation
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Notation, continued



yI p xq x

c

x

yp xq,

we will writeyI x

c

x

y.


Notation
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Notation, continued

This can make some differential equations confusing. Inthe ODE yI y, there is nothing to indicate what ydepends on (however you can deduce that y is thedependent variable, since we take a derivative).

Also, at our notational convenience, we will switchbetween Newtons notation and Leibnizs notation:

yI dy

dt, yP

d2y

dt2, . . . ,yp n q

dny

dtn.

When the derivative is with respect time, we might alsowrite 9y yI or :y yP .



Notation
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Notation, continued



yI dy

dt, yP

d2y

dt2, . . . ,yp n q

dny

dtn.




Notation
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Notation, continued



yI dy

dt

, yP d2y

dt2, . . . ,yp n q

dny

dtn.




Motivational Example
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Motivation

Lets look at a simple physical example of wheredifferential equations play a role: Newtonian motion.

First we recall two laws that Newton came up with:

Newtons Law of Gravity

Fgrav Gm1m2

r2

Newtons 2nd Law

F m a.

Also recall that if xp tq is the position of an object withrespect to time, then :xp tq a, the acceleration.



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Motivation




Fgrav Gm1m2

r2

Newtons 2nd Law

F m a.




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Motivation




Fgrav Gm1m2

r2

Newtons 2nd Law

F m a.




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Motivation




Fgrav Gm1m2

r2

Newtons 2nd Law

F m a.




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Motivation




Fgrav Gm1m2

r2

Newtons 2nd Law

F m a.






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Motivation

So if gravity is the only force acting on an object, then wemay equate Newtons two formula to find

m1

a

Gm1m2

r2.

Making obvious cancellations and substituting :xp tq a,we get

:

xp

tq

G

m2

r2.

On the surface of the earth, the number on the right isawful close to 9.8 m/s, which well just call g.





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Motivation


m1

a

G

m1m2

r2.


:

xp

tq

G

m2

r2.





M
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Motivation


m1

a

G

m1m2

r2.


:

xp

tq

G

m2

r2.





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Motivation

This is an easy example to quickly integrate (twice), andfind that

xp

tq

g

2 t2

v0t

x0.

We could make the model more sophisticated by addingin wind resistance, which acts proportionally againstvelocity:

m1 :xp tq m1g k 9xp tq .




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Motivation

This is an easy example to quickly integrate (twice), andfind that

xp

tq

g

2 t2

v0t

x0.

We could make the model more sophisticated by addingin wind resistance, which acts proportionally againstvelocity:

m1 :xp tq m1g k 9xp tq .
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Documents

An Introduction to Differential Equations