an introduction to differential equations...morris tenenbaum and harry pollard ordinary di erential...
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An Introduction to Differential Equations
An Introduction to Differential Equations
Colin Carroll
August 24, 2010
An Introduction to Differential Equations
Differential Equations
Ordinary things
Awesome things
mat
h 21
1
brea
kfas
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hugs
dinosaurs
jet packs
y’ = y
lightpurple
darkpurple
partial differential
equations
decaf
regular-strengthtylenol
the humaniti
es
the mediocre
wall of china
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:[email protected]
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:[email protected]
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:[email protected]
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:[email protected]
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.