An Introduction to Differential Equations
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An Introduction to Differential Equations An Introduction to Differential Equations Colin Carroll August 24, 2010 An Introduction to Differential Equations Awesome things Differential Equations regu lar -s tr tylen ength ol st y’ = y ial nt re fe s dif ion ial at rt equ pa diocre the me china wall of rs br ea kfa hu sau acks jet p 211 the es iti man math deca f Ordinary things din o dark purple hugs light purple An Introduction to Differential Equations 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. 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 by appointment. 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 by appointment. 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 by appointment. 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 by appointment. Course Webpage: http://math.rice.edu/ cc11 An Introduction to Differential Equations Syllabus Textbooks Syllabus- Textbooks Textbook : John Polking, Albert Boggess, David Arnold Differential Equations, Prentice Hall, 2nd Ed. Supplementary References: George Simmons, Stephen Krantz Differential Equations, McGraw Hill, Walter Rudin Student Series in Advanced Mathematics. Morris Tenenbaum and Harry Pollard Ordinary Differential Equations, Dover. An Introduction to Differential Equations Syllabus Textbooks Syllabus- Textbooks Textbook : John Polking, Albert Boggess, David Arnold Differential Equations, Prentice Hall, 2nd Ed. Supplementary References: George Simmons, Stephen Krantz Differential Equations, McGraw Hill, Walter Rudin Student Series in Advanced Mathematics. Morris Tenenbaum and Harry Pollard Ordinary Differential Equations, Dover. An Introduction to Differential Equations Syllabus Textbooks Syllabus- Textbooks Textbook : John Polking, Albert Boggess, David Arnold Differential Equations, Prentice Hall, 2nd Ed. Supplementary References: George Simmons, Stephen Krantz Differential Equations, McGraw Hill, Walter Rudin Student Series in Advanced Mathematics. Morris Tenenbaum and Harry Pollard Ordinary Differential Equations, Dover. An Introduction to Differential Equations Syllabus Textbooks Syllabus- Textbooks Textbook : John Polking, Albert Boggess, David Arnold Differential Equations, Prentice Hall, 2nd Ed. Supplementary References: George Simmons, Stephen Krantz Differential Equations, McGraw Hill, Walter Rudin Student Series in Advanced Mathematics. Morris Tenenbaum and Harry Pollard Ordinary 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 worth approximately: 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 a disability receive fair and equal treatment in this course. If you have a documented disability that requires academic adjustments or accommodation, please speak with me during the first week of class. All discussions will remain confidential. Students with disabilities will also need to contact Disability Support 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, or hopefully even a calculator. However, I plan on holding (approximately) two “intro to matlab” sessions during the semester. These will be helpful in checking work and likely if you 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 an equation that can be written in the form f x, y px q, y 1 px q, y 2 px q, . . . , y pnq px q 0. In this class, you will be asked to “solve” a differential equation, by which we mean find a function y px q that satisfies 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 an equation that can be written in the form f x, y px q, y 1 px q, y 2 px q, . . . , y pnq px q 0. In this class, you will be asked to “solve” a differential equation, by which we mean find a function y px q that satisfies 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 an equation that can be written in the form f x, y px q, y 1 px q, y 2 px q, . . . , y pnq px q 0. In this class, you will be asked to “solve” a differential equation, by which we mean find a function y px q that satisfies the above equation. This is unhelpful. Examples will help. An Introduction to Differential Equations Differential Equations Examples Example Solve the ODE y1 3x 2. From calculus we can calculate » y 1 dx ñy » 3x 2 dx x3 It doesn’t get any better than this. C. An Introduction to Differential Equations Differential Equations Examples Example Solve the ODE y1 3x 2. From calculus we can calculate » y 1 dx ñy » 3x 2 dx x3 It doesn’t get any better than this. C. An Introduction to Differential Equations Differential Equations Examples Example Solve the ODE y1 3x 2. From calculus we can calculate » y 1 dx ñy » 3x 2 dx x3 It doesn’t get any better than this. C. An Introduction to Differential Equations Differential Equations Examples Harder examples. What about solving the ODE y 1 y ? We cannot just integrate this, but there is a quick way to solve this. Similarly the differential equation y 2 y 0 looks fairly simple, but it will take most of the semester before we can solve it. We’ll be happy just verifying the solution for now. An Introduction to Differential Equations Differential Equations Examples Harder examples. What about solving the ODE y 1 y ? We cannot just integrate this, but there is a quick way to solve this. Similarly the differential equation y 2 y 0 looks fairly simple, but it will take most of the semester before we can solve it. We’ll be happy just verifying the solution for now. An Introduction to Differential Equations Differential Equations Examples Harder examples. What about solving the ODE y 1 y ? We cannot just integrate this, but there is a quick way to solve this. y Ae x . Similarly the differential equation y 2 y 0 looks fairly simple, but it will take most of the semester before we can solve it. We’ll be happy just verifying the solution for now. An Introduction to Differential Equations Differential Equations Examples Harder examples. What about solving the ODE y 1 y ? We cannot just integrate this, but there is a quick way to solve this. y Ae x . Similarly the differential equation y 2 y 0 looks fairly simple, but it will take most of the semester before we can solve it. We’ll be happy just verifying the solution for now. 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 y 2 y 0. Certainly y 1 A sin x B cos x. So y 2 A cos x B sin x. Then y2 y B sin x solves pA cos x B sin x q pA cos x as desired. B sin x q 0, An Introduction to Differential Equations Differential Equations Solutions Verifying Solutions We wish to show that y A cos x y 2 y 0. Certainly y 1 A sin x B cos x. So y 2 A cos x B sin x. Then y2 y B sin x solves pA cos x B sin x q pA cos x as desired. B sin x q 0, An Introduction to Differential Equations Differential Equations Solutions Verifying Solutions We wish to show that y A cos x y 2 y 0. Certainly y 1 A sin x B cos x. So y 2 A cos x B sin x. Then y2 y B sin x solves pA cos x B sin x q pA cos x as desired. B sin x q 0, An Introduction to Differential Equations Differential Equations Solutions Verifying Solutions We wish to show that y A cos x y 2 y 0. Certainly y 1 A sin x B cos x. So y 2 A cos x B sin x. Then y2 y B sin x solves pA cos x B sin x q pA cos x as desired. B sin x q 0, An Introduction to Differential Equations Differential Equations Solutions The Nature of Solutions Our intuition from calculus tells us that whatever we mean by “general solution”, it will not be unique, because of constants of integration. Indeed, by general solution, we mean writing down every solution to a differential equation- for an equation of order n, this will typically mean n constants of integration. We are also often concerned about a particular solution to an ODE. In this case, we will write down a differential equation 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 we mean by “general solution”, it will not be unique, because of constants of integration. Indeed, by general solution, we mean writing down every solution to a differential equation- for an equation of order n, this will typically mean n constants of integration. We are also often concerned about a particular solution to an ODE. In this case, we will write down a differential equation 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 we mean by “general solution”, it will not be unique, because of constants of integration. Indeed, by general solution, we mean writing down every solution to a differential equation- for an equation of order n, this will typically mean n constants of integration. We are also often concerned about a particular solution to an ODE. In this case, we will write down a differential equation as well as initial conditions. An Introduction to Differential Equations Differential Equations Solutions An example We will investigate the ODE x1 x sin t 2te cos t , with initial conditions x p0q 1. It turns out that a general solution to the ODE is x pt q pt 2 C qe cos t . Plugging in the initial condition gives us the particular solution x pt q pt 2 e qe cos t . An Introduction to Differential Equations Differential Equations Solutions An example We will investigate the ODE x1 x sin t 2te cos t , with initial conditions x p0q 1. It turns out that a general solution to the ODE is x pt q pt 2 C qe cos t . Plugging in the initial condition gives us the particular solution x pt q pt 2 e qe cos t . An Introduction to Differential Equations Differential Equations Solutions An example We will investigate the ODE x1 x sin t 2te cos t , with initial conditions x p0q 1. It turns out that a general solution to the ODE is x pt q pt 2 C qe cos t . Plugging in the initial condition gives us the particular solution x pt q pt 2 e qe cos t . An Introduction to Differential Equations Differential Equations Notation Some Notes on Notation Notation in differential equations can quickly become a mess. I try to follow fairly standard practices. The standard practices are sometimes confusing, but I would encourage you to emulate the notation used. If you still wish to use your own on a graded assignment please make your notation clear! Some general rules: we will usually use x or t as the independent variable, and y as the dependent variable. Unfortunately, the second choice for dependent variable is often x. An Introduction to Differential Equations Differential Equations Notation Some Notes on Notation Notation in differential equations can quickly become a mess. I try to follow fairly standard practices. The standard practices are sometimes confusing, but I would encourage you to emulate the notation used. If you still wish to use your own on a graded assignment please make your notation clear! Some general rules: we will usually use x or t as the independent variable, and y as the dependent variable. Unfortunately, the second choice for dependent variable is often x. An Introduction to Differential Equations Differential Equations Notation Some Notes on Notation Notation in differential equations can quickly become a mess. I try to follow fairly standard practices. The standard practices are sometimes confusing, but I would encourage you to emulate the notation used. If you still wish to use your own on a graded assignment please make your notation clear! Some general rules: we will usually use x or t as the independent variable, and y as the dependent variable. Unfortunately, the second choice for dependent variable is often x. An Introduction to Differential Equations Differential Equations Notation Notation, continued As above, we will usually suppress the dependence of one variable on another. That is to say, rather than write ?x y px q , ?x 1 yx . y y 1 px qx we will write An Introduction to Differential Equations Differential Equations Notation Notation, continued As above, we will usually suppress the dependence of one variable on another. That is to say, rather than write ?x y px q , ?x 1 yx . y y 1 px qx we will write An Introduction to Differential Equations Differential Equations Notation Notation, continued As above, we will usually suppress the dependence of one variable on another. That is to say, rather than write ?x y px q , ?x 1 yx . y y 1 px qx we will write An Introduction to Differential Equations Differential Equations Notation Notation, continued This can make some differential equations confusing. In the ODE y 1 y , there is nothing to indicate what y depends on (however you can deduce that y is the dependent variable, since we take a derivative). Also, at our notational convenience, we will switch between Newton’s notation and Leibniz’s notation: y1 dy , dt y2 2 ddty2 , . . . , y pn q n ddt yn . When the derivative is with respect time, we might also write y9 y 1 or y: y 2 . An Introduction to Differential Equations Differential Equations Notation Notation, continued This can make some differential equations confusing. In the ODE y 1 y , there is nothing to indicate what y depends on (however you can deduce that y is the dependent variable, since we take a derivative). Also, at our notational convenience, we will switch between Newton’s notation and Leibniz’s notation: y1 dy , dt y2 2 ddty2 , . . . , y pn q n ddt yn . When the derivative is with respect time, we might also write y9 y 1 or y: y 2 . An Introduction to Differential Equations Differential Equations Notation Notation, continued This can make some differential equations confusing. In the ODE y 1 y , there is nothing to indicate what y depends on (however you can deduce that y is the dependent variable, since we take a derivative). Also, at our notational convenience, we will switch between Newton’s notation and Leibniz’s notation: y1 dy , dt y2 2 ddty2 , . . . , y pn q n ddt yn . When the derivative is with respect time, we might also write y9 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 where differential equations play a role: Newtonian motion. First we recall two laws that Newton came up with: Newton’s Law of Gravity Fgrav m1 m2 r2 G m a. Newton’s 2nd Law F Also recall that if x pt q is the position of an object with respect to time, then x:pt q a, the acceleration. An Introduction to Differential Equations Differential Equations Motivational Example Motivation Let’s look at a simple physical example of where differential equations play a role: Newtonian motion. First we recall two laws that Newton came up with: Newton’s Law of Gravity Fgrav m1 m2 r2 G m a. Newton’s 2nd Law F Also recall that if x pt q is the position of an object with respect to time, then x:pt q a, the acceleration. An Introduction to Differential Equations Differential Equations Motivational Example Motivation Let’s look at a simple physical example of where differential equations play a role: Newtonian motion. First we recall two laws that Newton came up with: Newton’s Law of Gravity Fgrav m1 m2 r2 G m a. Newton’s 2nd Law F Also recall that if x pt q is the position of an object with respect to time, then x:pt q a, the acceleration. An Introduction to Differential Equations Differential Equations Motivational Example Motivation Let’s look at a simple physical example of where differential equations play a role: Newtonian motion. First we recall two laws that Newton came up with: Newton’s Law of Gravity Fgrav m1 m2 r2 G m a. Newton’s 2nd Law F Also recall that if x pt q is the position of an object with respect to time, then x:pt q a, the acceleration. An Introduction to Differential Equations Differential Equations Motivational Example Motivation Let’s look at a simple physical example of where differential equations play a role: Newtonian motion. First we recall two laws that Newton came up with: Newton’s Law of Gravity Fgrav m1 m2 r2 G m a. Newton’s 2nd Law F Also recall that if x pt q is the position of an object with respect to time, then x:pt q 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 we may equate Newton’s two formula to find m1 a G m1 m2 . r2 Making obvious cancellations and substituting x:pt q a, we get m2 x:pt q G 2 . r On the surface of the earth, the number on the right is awful 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 we may equate Newton’s two formula to find m1 a G m1 m2 . r2 Making obvious cancellations and substituting x:pt q a, we get m2 x:pt q G 2 . r On the surface of the earth, the number on the right is awful 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 we may equate Newton’s two formula to find m1 a G m1 m2 . r2 Making obvious cancellations and substituting x:pt q a, we get m2 x:pt q G 2 . r On the surface of the earth, the number on the right is awful 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), and find that g x p t q t 2 v0 t x0 . 2 We could make the model more sophisticated by adding in wind resistance, which acts proportionally against velocity: m1 x:pt q m1 g k x pt q. 9 An Introduction to Differential Equations Differential Equations Motivational Example Motivation This is an easy example to quickly integrate (twice), and find that g x p t q t 2 v0 t x0 . 2 We could make the model more sophisticated by adding in wind resistance, which acts proportionally against velocity: m1 x:pt q m1 g k x pt q. 9
