an introduction to differential equations...morris tenenbaum and harry pollard ordinary di erential...

An Introduction to Differential Equations

An Introduction to Differential Equations

Colin Carroll

August 24, 2010

An Introduction to Differential Equations

Differential Equations

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An Introduction to Differential Equations

Syllabus

Basic Info

Syllabus- Are You In the Right Room?

MATH 211 - ORDINARY DIFFERENTIALEQUATIONS AND LINEAR ALGEBRA

FALL 2010TR 9:25 - 10:40am, HBH427.

An Introduction to Differential Equations

Syllabus

Basic Info

Syllabus- How To Get In Touch

Instructor: Colin Carroll

Contact Info: Office: HB 447, Phone: x4598, E-mail:colin.carroll@rice.edu

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

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

An Introduction to Differential Equations

Syllabus

Basic Info

Syllabus- How To Get In Touch

Instructor: Colin Carroll

Contact Info: Office: HB 447, Phone: x4598, E-mail:colin.carroll@rice.edu

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

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

An Introduction to Differential Equations

Syllabus

Basic Info

Syllabus- How To Get In Touch

Instructor: Colin Carroll

Contact Info: Office: HB 447, Phone: x4598, E-mail:colin.carroll@rice.edu

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

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

An Introduction to Differential Equations

Syllabus

Basic Info

Syllabus- How To Get In Touch

Instructor: Colin Carroll

Contact Info: Office: HB 447, Phone: x4598, E-mail:colin.carroll@rice.edu

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

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

An Introduction to Differential Equations

Syllabus

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

An Introduction to Differential Equations

Syllabus

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

An Introduction to Differential Equations

Syllabus

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

An Introduction to Differential Equations

Syllabus

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

An Introduction to Differential Equations

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!

An Introduction to Differential Equations

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!

An Introduction to Differential Equations

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!

An Introduction to Differential Equations

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!

An Introduction to Differential Equations

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!

An Introduction to Differential Equations

Syllabus

Grading

Syllabus- Exams

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

Exams from the summer are available on my website.

An Introduction to Differential Equations

Syllabus

Grading

Syllabus- Exams

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

Exams from the summer are available on my website.

An Introduction to Differential Equations

Syllabus

Grading

Grades.

Grades will be based on homeworks and exams, and worthapproximately:

Homeworks: 15 %

Midterm Exam I: 20 %

Midterm Exam II: 25 %

Final Exam: 40 %

An Introduction to Differential Equations

Syllabus

Disability Support

Syllabus- Disability Support

It is the policy of Rice University that any student with adisability 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.

An Introduction to Differential Equations

Syllabus

Important Dates

Syllabus- Important Dates

Tuesday, August 24: First class.

September 30-October 5: Midterm exam I

Tuesday, 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.

An Introduction to Differential Equations

Syllabus

A Note on Technology

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.

An Introduction to Differential Equations

Syllabus

A Note on Technology

Pause for questions, applause.

An Introduction to Differential Equations

Differential Equations

Introduction

What is an Ordinary Differential Equation?

An ordinary differential equation (also called an ODE, or”DiffEQ”, pronounced ”diffy-Q” by the cool kids) is anequation that can be written in the form

f�x , ypxq, y 1pxq, y 2pxq, . . . , y pnqpxq� � 0.

In this class, you will be asked to “solve” a differentialequation, by which we mean find a function ypxq thatsatisfies the above equation.

This is unhelpful. Examples will help.

An Introduction to Differential Equations

Differential Equations

Introduction

What is an Ordinary Differential Equation?

An ordinary differential equation (also called an ODE, or”DiffEQ”, pronounced ”diffy-Q” by the cool kids) is anequation that can be written in the form

f�x , ypxq, y 1pxq, y 2pxq, . . . , y pnqpxq� � 0.

In this class, you will be asked to “solve” a differentialequation, by which we mean find a function ypxq thatsatisfies the above equation.

This is unhelpful. Examples will help.

An Introduction to Differential Equations

Differential Equations

Introduction

What is an Ordinary Differential Equation?

An ordinary differential equation (also called an ODE, or”DiffEQ”, pronounced ”diffy-Q” by the cool kids) is anequation that can be written in the form

f�x , ypxq, y 1pxq, y 2pxq, . . . , y pnqpxq� � 0.

In this class, you will be asked to “solve” a differentialequation, by which we mean find a function ypxq thatsatisfies the above equation.

This is unhelpful. Examples will help.

An Introduction to Differential Equations

Differential Equations

Examples

Example

Solve the ODEy 1 � 3x2.

From calculus we can calculate»y 1 dx �

»3x2 dx

ñ y � x3 � C .

It doesn’t get any better than this.

An Introduction to Differential Equations

Differential Equations

Examples

Example

Solve the ODEy 1 � 3x2.

From calculus we can calculate»y 1 dx �

»3x2 dx

ñ y � x3 � C .

It doesn’t get any better than this.

An Introduction to Differential Equations

Differential Equations

Examples

Example

Solve the ODEy 1 � 3x2.

From calculus we can calculate»y 1 dx �

»3x2 dx

ñ y � x3 � C .

It doesn’t get any better than this.

An Introduction to Differential Equations

Differential Equations

Examples

Harder examples.

What about solving the ODE y 1 � y? We cannot justintegrate this, but there is a quick way to solve this.

y � Aex .

Similarly the differential equation y 2 � y � 0 looks fairlysimple, but it will take most of the semester before wecan solve it. We’ll be happy just verifying the solution fornow.

y � A cos x � B sin x .

An Introduction to Differential Equations

Differential Equations

Examples

Harder examples.

What about solving the ODE y 1 � y? We cannot justintegrate this, but there is a quick way to solve this.

y � Aex .

Similarly the differential equation y 2 � y � 0 looks fairlysimple, but it will take most of the semester before wecan solve it. We’ll be happy just verifying the solution fornow.

y � A cos x � B sin x .

An Introduction to Differential Equations

Differential Equations

Examples

Harder examples.

What about solving the ODE y 1 � y? We cannot justintegrate this, but there is a quick way to solve this.

y � Aex .

Similarly the differential equation y 2 � y � 0 looks fairlysimple, but it will take most of the semester before wecan solve it. We’ll be happy just verifying the solution fornow.

y � A cos x � B sin x .

An Introduction to Differential Equations

Differential Equations

Examples

Harder examples.

What about solving the ODE y 1 � y? We cannot justintegrate this, but there is a quick way to solve this.

y � Aex .

Similarly the differential equation y 2 � y � 0 looks fairlysimple, but it will take most of the semester before wecan solve it. We’ll be happy just verifying the solution fornow.

y � A cos x � B sin x .

An Introduction to Differential Equations

Differential Equations

Solutions

Verifying Solutions

We wish to show that y � A cos x � B sin x solvesy 2 � y � 0.

Certainly y 1 � �A sin x � B cos x .

So y 2 � �A cos x � B sin x .

Then

y 2 � y � p�A cos x � B sin xq � pA cos x � B sin xq � 0,

as desired.

An Introduction to Differential Equations

Differential Equations

Solutions

Verifying Solutions

We wish to show that y � A cos x � B sin x solvesy 2 � y � 0.

Certainly y 1 � �A sin x � B cos x .

So y 2 � �A cos x � B sin x .

Then

y 2 � y � p�A cos x � B sin xq � pA cos x � B sin xq � 0,

as desired.

An Introduction to Differential Equations

Differential Equations

Solutions

Verifying Solutions

We wish to show that y � A cos x � B sin x solvesy 2 � y � 0.

Certainly y 1 � �A sin x � B cos x .

So y 2 � �A cos x � B sin x .

Then

y 2 � y � p�A cos x � B sin xq � pA cos x � B sin xq � 0,

as desired.

An Introduction to Differential Equations

Differential Equations

Solutions

Verifying Solutions

We wish to show that y � A cos x � B sin x solvesy 2 � y � 0.

Certainly y 1 � �A sin x � B cos x .

So y 2 � �A cos x � B sin x .

Then

y 2 � y � p�A cos x � B sin xq � pA cos x � B sin xq � 0,

as desired.

An Introduction to Differential Equations

Differential Equations

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.

An Introduction to Differential Equations

Differential Equations

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.

An Introduction to Differential Equations

Differential Equations

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.

An Introduction to Differential Equations

Differential Equations

Solutions

An example

We will investigate the ODE

x 1 � x sin t � 2te� cos t , with initial conditions xp0q � 1.

It turns out that a general solution to the ODE is

xptq � pt2 � C qe� cos t .

Plugging in the initial condition gives us the particularsolution

xptq � pt2 � eqe� cos t .

An Introduction to Differential Equations

Differential Equations

Solutions

An example

We will investigate the ODE

x 1 � x sin t � 2te� cos t , with initial conditions xp0q � 1.

It turns out that a general solution to the ODE is

xptq � pt2 � C qe� cos t .

Plugging in the initial condition gives us the particularsolution

xptq � pt2 � eqe� cos t .

An Introduction to Differential Equations

Differential Equations

Solutions

An example

We will investigate the ODE

x 1 � x sin t � 2te� cos t , with initial conditions xp0q � 1.

It turns out that a general solution to the ODE is

xptq � pt2 � C qe� cos t .

Plugging in the initial condition gives us the particularsolution

xptq � pt2 � eqe� cos t .

An Introduction to Differential Equations

Differential Equations

Notation

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 .

An Introduction to Differential Equations

Differential Equations

Notation

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 .

An Introduction to Differential Equations

Differential Equations

Notation

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 .

An Introduction to Differential Equations

Differential Equations

Notation

Notation, continued

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

That is to say, rather than write

y 1pxqx �?

x

ypxq ,

we will write

y 1x �?

x

y.

An Introduction to Differential Equations

Differential Equations

Notation

Notation, continued

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

That is to say, rather than write

y 1pxqx �?

x

ypxq ,

we will write

y 1x �?

x

y.

An Introduction to Differential Equations

Differential Equations

Notation

Notation, continued

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

That is to say, rather than write

y 1pxqx �?

x

ypxq ,

we will write

y 1x �?

x

y.

An Introduction to Differential Equations

Differential Equations

Notation

Notation, continued

This can make some differential equations confusing. Inthe ODE y 1 � 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 Newton’s notation and Leibniz’s notation:

y 1 � dy

dt, y 2 � d2y

dt2, . . . , y pnq � dny

dtn.

When the derivative is with respect time, we might alsowrite 9y � y 1 or :y � y 2.

An Introduction to Differential Equations

Differential Equations

Notation

Notation, continued

This can make some differential equations confusing. Inthe ODE y 1 � 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 Newton’s notation and Leibniz’s notation:

y 1 � dy

dt, y 2 � d2y

dt2, . . . , y pnq � dny

dtn.

When the derivative is with respect time, we might alsowrite 9y � y 1 or :y � y 2.

An Introduction to Differential Equations

Differential Equations

Notation

Notation, continued

This can make some differential equations confusing. Inthe ODE y 1 � 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 Newton’s notation and Leibniz’s notation:

y 1 � dy

dt, y 2 � d2y

dt2, . . . , y pnq � dny

dtn.

When the derivative is with respect time, we might alsowrite 9y � y 1 or :y � y 2.

An Introduction to Differential Equations

Differential Equations

Motivational Example

Motivation

Let’s look at a simple physical example of wheredifferential equations play a role: Newtonian motion.

First we recall two laws that Newton came up with:

Newton’s Law of Gravity

Fgrav � Gm1m2

r2

Newton’s 2nd Law

F � m � a.

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

An Introduction to Differential Equations

Differential Equations

Motivational Example

Motivation

Let’s look at a simple physical example of wheredifferential equations play a role: Newtonian motion.

First we recall two laws that Newton came up with:

Newton’s Law of Gravity

Fgrav � Gm1m2

r2

Newton’s 2nd Law

F � m � a.

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

An Introduction to Differential Equations

Differential Equations

Motivational Example

Motivation

Let’s look at a simple physical example of wheredifferential equations play a role: Newtonian motion.

First we recall two laws that Newton came up with:

Newton’s Law of Gravity

Fgrav � Gm1m2

r2

Newton’s 2nd Law

F � m � a.

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

An Introduction to Differential Equations

Differential Equations

Motivational Example

Motivation

Let’s look at a simple physical example of wheredifferential equations play a role: Newtonian motion.

First we recall two laws that Newton came up with:

Newton’s Law of Gravity

Fgrav � Gm1m2

r2

Newton’s 2nd Law

F � m � a.

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

An Introduction to Differential Equations

Differential Equations

Motivational Example

Motivation

Let’s look at a simple physical example of wheredifferential equations play a role: Newtonian motion.

First we recall two laws that Newton came up with:

Newton’s Law of Gravity

Fgrav � Gm1m2

r2

Newton’s 2nd Law

F � m � a.

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

An Introduction to Differential Equations

Differential Equations

Motivational Example

Motivation

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

m1 � a � Gm1m2

r 2.

Making obvious cancellations and substituting :xptq � a,we get

:xptq � Gm2

r 2.

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

An Introduction to Differential Equations

Differential Equations

Motivational Example

Motivation

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

m1 � a � Gm1m2

r 2.

Making obvious cancellations and substituting :xptq � a,we get

:xptq � Gm2

r 2.

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

An Introduction to Differential Equations

Differential Equations

Motivational Example

Motivation

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

m1 � a � Gm1m2

r 2.

Making obvious cancellations and substituting :xptq � a,we get

:xptq � Gm2

r 2.

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

An Introduction to Differential Equations

Differential Equations

Motivational Example

Motivation

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

xptq � g

2t2 � v0t � x0.

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

m1:xptq � m1g � k 9xptq.

An Introduction to Differential Equations

Differential Equations

Motivational Example

Motivation

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

xptq � g

2t2 � v0t � x0.

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

m1:xptq � m1g � k 9xptq.