# The integrating factor method (Sect. 2.1).

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 The integrating factor method (Sect. 2.1). Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Variable coefficients. Read: The direction field. 2 in Section 1.1 in the Textbook. See direction field plotters in Internet. For example, see: dfield/dfpp.html This link is given in our class webpage. Overview of differential equations. Definition A differential equation is an equation, where the unknown is a function, and both the function and its derivative appear in the equation. Remark: There are two main types of differential equations: Ordinary Differential Equations (ODE): Derivatives with respect to only one variable appear in the equation. : Newton s second law of motion: m a = F. Partial differential Equations (PDE): Partial derivatives of two or more variables appear in the equation. : The wave equation for sound propagation in air.

2 Overview of differential equations. Newton s second law of motion is an ODE: The unknown is x(t), the particle position as function of time t and the equation is d 2 dt 2 x(t) = 1 F(t, x(t)), m with m the particle mass and F the force acting on the particle. The wave equation is a PDE: The unknown is u(t, x), a function that depends on two variables, and the equation is u(t, x) = v u(t, x), t2 x 2 with v the wave speed. Sound propagation in air is described by a wave equation, where u represents the air pressure. Overview of differential equations. Remark: Differential equations are a central part in a physical description of nature: Classical Mechanics: Newton s second law of motion. (ODE) Lagrange s equations. (ODE) Electromagnetism: Maxwell s equations. (PDE) Quantum Mechanics: Schrödinger s equation. (PDE) General Relativity: Einstein equation. (PDE) Quantum Electrodynamics: The equations of QED. (PDE).

3 The integrating factor method (Sect. 2.1). Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Variable coefficients. Linear Ordinary Differential Equations Remark: Given a function y : R R, we use the notation y (t) = dy dt (t). Definition Given a function f : R 2 R, a first order ODE in the unknown function y : R R is the equation y (t) = f (t, y(t)). The first order ODE above is called linear iff there exist functions a, b : R R such that f (t, y) = a(t) y + b(t). That is, f is linear on its argument y, hence a first order linear ODE is given by y (t) = a(t) y(t) + b(t).

4 Linear Ordinary Differential Equations A first order linear ODE is given by y (t) = 2 y(t) + 3. In this case function a(t) = 2 and b(t) = 3. Since these function do not depend on t, the equation above is called of constant coefficients. A first order linear ODE is given by y (t) = 2 t y(t) + 4t. In this case function a(t) = 2/t and b(t) = 4t. Since these functions depend on t, the equation above is called of variable coefficients. The integrating factor method (Sect. 2.1). Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Variable coefficients.

5 The integrating factor method. Remark: Solutions to first order linear ODE can be obtained using the integrating factor method. Theorem (Constant coefficients) Given constants a, b R with a 0, the linear differential equation y (t) = a y(t) + b has infinitely many solutions, one for each value of c R, given by y(t) = c e at + b a. The integrating factor method. Proof: Multiply the differential equation y (t) + a y(t) = b by a non-zero function µ, that is, µ(t) ( y + ay ) = µ(t) b. Key idea: The non-zero function µ is called an integrating factor iff holds µ ( y + ay ) = ( µ y ). Not every function µ satisfies the equation above. Let us find what are the solutions µ of the equation above. Notice that µ ( y + ay ) = ( µ y ) µ y + µay = µ y + µ y ayµ = µ y aµ = µ µ (t) µ(t) = a.

6 The integrating factor method. Proof: Recall: µ (t) µ(t) = a. Therefore, [ ln ( µ(t) )] = a ln ( µ(t) ) = at + c0, µ(t) = e at+c 0 µ(t) = e at e c 0. Choosing the solution with c 0 = 0 we obtain µ(t) = e at. For that function µ holds that µ ( y + ay ) = ( µ y ). Therefore, multiplying the ODE y + ay = b by µ = e at we get (µy) = bµ ( e at y ) = be at e at y = be at dt + c y(t) e at = b a eat + c y(t) = c e at + b a. The integrating factor method. Find all functions y solution of the ODE y = 2y + 3. Solution: The ODE is y = ay + b with a = 2 and b = 3. The functions y(t) = ce at + b, with c R, are solutions. a We conclude that the ODE has infinitely many solutions, given by y c > 0 y(t) = c e 2t 3 2, c R. Since we did one integration, it is reasonable that the solution contains a constant of integration, c R. 0 3/2 t c = 0 c < 0 Verification: c e 2t = y + (3/2), so 2c e 2t = y, therefore we conclude that y satisfies the ODE y = 2y + 3.

7 The integrating factor method (Sect. 2.1). Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Variable coefficients. The Initial Value Problem. Definition The Initial Value Problem (IVP) for a linear ODE is the following: Given functions a, b : R R and constants t 0, y 0 R, find a solution y : R R of the problem y = a(t) y + b(t), y(t 0 ) = y 0. Remark: The initial condition selects one solution of the ODE. Theorem (Constant coefficients) Given constants a, b, t 0, y 0 R, with a 0, the initial value problem y = ay + b, y(t 0 ) = y 0 has the unique solution y(t) = ( y 0 b ) e a(t t0) + b a a.

8 The Initial Value Problem. Find the solution to the initial value problem y = 2y + 3, y(0) = 1. Solution: Every solution of the ODE above is given by y(t) = c e 2t 3 2, c R. The initial condition y(0) = 1 selects only one solution: 1 = y(0) = c 3 2 c = 5 2. We conclude that y(t) = 5 2 e2t 3 2. The integrating factor method (Sect. 2.1). Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Variable coefficients.

9 The integrating factor method. Theorem (Variable coefficients) Given continuous functions a, b : R R and given constants t 0, y 0 R, the IVP y = a(t)y + b(t) y(t 0 ) = y 0 has the unique solution y(t) = 1 [ t ] y 0 + µ(s)b(s)ds, µ(t) t 0 where the integrating factor function is given by µ(t) = e A(t), A(t) = t Remark: See the proof in the Lecture Notes. t 0 a(s)ds. The integrating factor method. Find the solution y to the IVP t y + 2y = 4t 2, y(1) = 2. Solution: We first express the ODE as in the Theorem above, y = 2 t y + 4t. Therefore, a(t) = 2 t and b(t) = 4t, and also t 0 = 1 and y 0 = 2. We first compute the integrating factor function µ = e A(t), where A(t) = t t 0 a(s) ds = t 1 2 s ds = 2[ ln(t) ln(1) ] A(t) = 2 ln(t) = ln(t 2 ) e A(t) = t 2. We conclude that µ(t) = t 2.

10 The integrating factor method. Find the solution y to the IVP t y + 2y = 4t 2, y(1) = 2. Solution: The integrating factor is µ(t) = t 2. Hence, t 2( y + 2 ) t y = t 2 (4t) t 2 y + 2t y = 4t 3 ( t 2 y ) = 4t 3 t 2 y = t 4 + c y = t 2 + c t 2. The initial condition implies 2 = y(1) = 1 + c, that is, c = 1. We conclude that y(t) = t t 2.

### ORDINARY DIFFERENTIAL EQUATIONS

ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. SEPTEMBER 4, 25 Summary. This is an introduction to ordinary differential equations.

### A First Course in Elementary Differential Equations. Marcel B. Finan Arkansas Tech University c All Rights Reserved

A First Course in Elementary Differential Equations Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 Contents 1 Basic Terminology 4 2 Qualitative Analysis: Direction Field of y = f(t, y)

### 22 Matrix exponent. Equal eigenvalues

22 Matrix exponent. Equal eigenvalues 22. Matrix exponent Consider a first order differential equation of the form y = ay, a R, with the initial condition y) = y. Of course, we know that the solution to

### LS.6 Solution Matrices

LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions

### General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1

A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1 Dr. John Ehrke Department of Mathematics Fall 2012 Questions

### Name: Exam 1. y = 10 t 2. y(0) = 3

Instructions. Answer each of the questions on your own paper. Be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

### (4.8) Solving Systems of Linear DEs by Elimination

INTRODUCTION: (4.8) Solving Systems of Linear DEs by Elimination Simultaneous dinary differential equations involve two me equations that contain derivatives of two me dependent variables the unknown functions

### First Order Non-Linear Equations

First Order Non-Linear Equations We will briefly consider non-linear equations. In general, these may be much more difficult to solve than linear equations, but in some cases we will still be able to solve

### Harmonic Oscillator Physics

Physics 34 Lecture 9 Harmonic Oscillator Physics Lecture 9 Physics 34 Quantum Mechanics I Friday, February th, 00 For the harmonic oscillator potential in the time-independent Schrödinger equation: d ψx

### Sample Solutions of Assignment 5 for MAT3270B: Section 3.4. In each of problems find the general solution of the given differential equation

Sample Solutions of Assignment 5 for MAT370B: 3.4-3.9 Section 3.4 In each of problems find the general solution of the given differential equation 7. y y + y = 0 1. 4y + 9y = 0 14. 9y + 9y 4y = 0 15. y

### Solving First Order Linear Equations

Solving First Order Linear Equations Today we will discuss how to solve a first order linear equation. Our technique will require close familiarity with the product and chain rules for derivatives, so

### Chapter 4: Higher-Order Differential Equations Part 3

Chapter 4: Higher-Order Differential Equations Part 3 王奕翔 Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw October 24, 2013 1 / 17 王奕翔 DE Lecture 7 1 Solving Systems of

### Homework #2 Solutions

MAT Spring Problems Section.:, 8,, 4, 8 Section.5:,,, 4,, 6 Extra Problem # Homework # Solutions... Sketch likely solution curves through the given slope field for dy dx = x + y...8. Sketch likely solution

### ARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida

ARBITRAGE-FREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic

### Math 22B, Homework #8 1. y 5y + 6y = 2e t

Math 22B, Homework #8 3.7 Problem # We find a particular olution of the ODE y 5y + 6y 2e t uing the method of variation of parameter and then verify the olution uing the method of undetermined coefficient.

### Student name: Earlham College. Fall 2011 December 15, 2011

Student name: Earlham College MATH 320: Differential Equations Final exam - In class part Fall 2011 December 15, 2011 Instructions: This is a regular closed-book test, and is to be taken without the use

### Class Meeting # 1: Introduction to PDEs

MATH 18.152 COURSE NOTES - CLASS MEETING # 1 18.152 Introduction to PDEs, Fall 2011 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u = u(x

### 1 Gaussian Elimination

Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 Gauss-Jordan reduction and the Reduced

### Reduced echelon form: Add the following conditions to conditions 1, 2, and 3 above:

Section 1.2: Row Reduction and Echelon Forms Echelon form (or row echelon form): 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry (i.e. left most nonzero entry) of a row is in

### The Variational Principle, from Fermat to Feynman. Nick Sheridan

The Variational Principle, from Fermat to Feynman Nick Sheridan 1662: Fermat points out that light always follows the path between two points that takes least time. This explains the laws of reflection

### MATH 425, PRACTICE FINAL EXAM SOLUTIONS.

MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator

### Math 2280 Section 002 [SPRING 2013] 1

Math 2280 Section 002 [SPRING 2013] 1 Today well learn about a method for solving systems of differential equations, the method of elimination, that is very similar to the elimination methods we learned

### Lecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 10

Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 10 Boundary Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction

### 5 Homogeneous systems

5 Homogeneous systems Definition: A homogeneous (ho-mo-jeen -i-us) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x +... + a n x n =.. a m

### Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product

Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product Geometrical definition Properties Expression in components. Definition in components Properties Geometrical expression.

### Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem

Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Gagan Deep Singh Assistant Vice President Genpact Smart Decision Services Financial

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### A First Course in Elementary Differential Equations: Problems and Solutions. Marcel B. Finan Arkansas Tech University c All Rights Reserved

A First Course in Elementary Differential Equations: Problems and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 Contents 1 Basic Terminology 4 2 Qualitative Analysis: Direction

### 9. Particular Solutions of Non-homogeneous second order equations Undetermined Coefficients

September 29, 201 9-1 9. Particular Solutions of Non-homogeneous second order equations Undetermined Coefficients We have seen that in order to find the general solution to the second order differential

### 5.1. Systems of Linear Equations. Linear Systems Substitution Method Elimination Method Special Systems

5.1 Systems of Linear Equations Linear Systems Substitution Method Elimination Method Special Systems 5.1-1 Linear Systems The possible graphs of a linear system in two unknowns are as follows. 1. The

### Lecture 14: Section 3.3

Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in

### 9 Multiplication of Vectors: The Scalar or Dot Product

Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation

### Problems for Quiz 14

Problems for Quiz 14 Math 3. Spring, 7. 1. Consider the initial value problem (IVP defined by the partial differential equation (PDE u t = u xx u x + u, < x < 1, t > (1 with boundary conditions and initial

### CHAPTER 3. Fourier Series

`A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 4 A COLLECTION OF HANDOUTS ON PARTIAL DIFFERENTIAL

### MATH 31B: MIDTERM 1 REVIEW. 1. Inverses. yx 3y = 1. x = 1 + 3y y 4( 1) + 32 = 1

MATH 3B: MIDTERM REVIEW JOE HUGHES. Inverses. Let f() = 3. Find the inverse g() for f. Solution: Setting y = ( 3) and solving for gives and g() = +3. y 3y = = + 3y y. Let f() = 4 + 3. Find a domain on

### Dot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product

Dot product and vector projections (Sect. 12.3) Two definitions for the dot product. Geometric definition of dot product. Orthogonal vectors. Dot product and orthogonal projections. Properties of the dot

### Scientific Computing: An Introductory Survey

Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign

### Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t.

LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing

### Lecture 3: Fourier Series: pointwise and uniform convergence.

Lecture 3: Fourier Series: pointwise and uniform convergence. 1. Introduction. At the end of the second lecture we saw that we had for each function f L ([, π]) a Fourier series f a + (a k cos kx + b k

### 7. Continuously Varying Interest Rates

7. Continuously Varying Interest Rates 7.1 The Continuous Varying Interest Rate Formula. Suppose that interest is continuously compounded with a rate which is changing in time. Let the present time be

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### 20 Applications of Fourier transform to differential equations

20 Applications of Fourier transform to differential equations Now I did all the preparatory work to be able to apply the Fourier transform to differential equations. The key property that is at use here

### 1. First-order Ordinary Differential Equations

Advanced Engineering Mathematics 1. First-order ODEs 1 1. First-order Ordinary Differential Equations 1.1 Basic concept and ideas 1.2 Geometrical meaning of direction fields 1.3 Separable differential

### Fourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks

Chapter 3 Fourier Series 3.1 Some Properties of Functions 3.1.1 Goal We review some results about functions which play an important role in the development of the theory of Fourier series. These results

### CHAPTER 2. Eigenvalue Problems (EVP s) for ODE s

A SERIES OF CLASS NOTES FOR 005-006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 4 A COLLECTION OF HANDOUTS ON PARTIAL DIFFERENTIAL EQUATIONS

### Lecture Notes Math 250: Ordinary Differential Equations

Lecture Notes Math 250: Ordinary Differential Equations Wen Shen 20 NB! These notes are used by myself. They are provided to students as a supplement to the textbook. They can not substitute the textbook.

### Nonlinear Systems of Ordinary Differential Equations

Differential Equations Massoud Malek Nonlinear Systems of Ordinary Differential Equations Dynamical System. A dynamical system has a state determined by a collection of real numbers, or more generally

### An Introduction to Separation of Variables with Fourier Series Math 391w, Spring 2010 Tim McCrossen Professor Haessig

An Introduction to Separation of Variables with Fourier Series Math 391w, Spring 2010 Tim McCrossen Professor Haessig Abstract: This paper aims to give students who have not yet taken a course in partial

### System of First Order Differential Equations

CHAPTER System of First Order Differential Equations In this chapter, we will discuss system of first order differential equations. There are many applications that involving find several unknown functions

### SECOND-ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS

L SECOND-ORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS SECOND-ORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS A second-order linear differential equation is one of the form d

### MATH 215/255 Fall 2014 Assignment 2

MATH 215/255 Fall 214 Assignment 2 1.4, Exact equations ([Braun s Section 1.9]), 1.6 Solutions to selected exercises can be found in [Lebl], starting from page 33. 1.4.8: Solve 1 x 2 + 1 y + xy = 3 with

### Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.

Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard

### Math 432 HW 2.6 Solutions

Assigned: 1-8, 11, 19, 25, 29, 32, 34, 39, and 40. Selected for Grading: 2, 7, 34, 39 Math 432 HW 2.6 Solutions Solutions: 1. Given: (y 4x 1) 2 dx dy = 0. Solving this for dy/dx gives dy/dx = (y 4x 1)

### From Einstein to Klein-Gordon Quantum Mechanics and Relativity

From Einstein to Klein-Gordon Quantum Mechanics and Relativity Aline Ribeiro Department of Mathematics University of Toronto March 24, 2002 Abstract We study the development from Einstein s relativistic

### Solutions to Linear First Order ODE s

First Order Linear Equations In the previous session we learned that a first order linear inhomogeneous ODE for the unknown function x = x(t), has the standard form x + p(t)x = q(t) () (To be precise we

### Consider the first-order wave equation with constant speed: u t + c u x = 0. It responds well to a change of variables: x ξ + η.

A First-order PDEs First-order partial differential equations can be tackled with the method of characteristics, a powerful tool which also reaches beyond first-order. We ll be looking primarily at equations

### A Second Course in Elementary Differential Equations: Problems and Solutions. Marcel B. Finan Arkansas Tech University c All Rights Reserved

A Second Course in Elementary Differential Equations: Problems and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved Contents 8 Calculus of Matrix-Valued Functions of a Real Variable

### Linear Systems. Singular and Nonsingular Matrices. Find x 1, x 2, x 3 such that the following three equations hold:

Linear Systems Example: Find x, x, x such that the following three equations hold: x + x + x = 4x + x + x = x + x + x = 6 We can write this using matrix-vector notation as 4 {{ A x x x {{ x = 6 {{ b General

### 1 Lecture 3: Operators in Quantum Mechanics

1 Lecture 3: Operators in Quantum Mechanics 1.1 Basic notions of operator algebra. In the previous lectures we have met operators: ˆx and ˆp = i h they are called fundamental operators. Many operators

### To give it a definition, an implicit function of x and y is simply any relationship that takes the form:

2 Implicit function theorems and applications 21 Implicit functions The implicit function theorem is one of the most useful single tools you ll meet this year After a while, it will be second nature to

### Pricing Barrier Options under Local Volatility

Abstract Pricing Barrier Options under Local Volatility Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 16 November 2002 We study pricing under the local volatility. Our research is mainly

### Introduction to Sturm-Liouville Theory

Introduction to Ryan C. Trinity University Partial Differential Equations April 10, 2012 Inner products with weight functions Suppose that w(x) is a nonnegative function on [a,b]. If f (x) and g(x) are

### Partial Fraction Decomposition for Inverse Laplace Transform

Partial Fraction Decomposition for Inverse Laplace Transform Usually partial fractions method starts with polynomial long division in order to represent a fraction as a sum of a polynomial and an another

### PHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Exam Solutions Dec. 13, 2004

PHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Exam Solutions Dec. 1, 2004 No materials allowed. If you can t remember a formula, ask and I might help. If you can t do one part of a problem,

### Methods for Finding Bases

Methods for Finding Bases Bases for the subspaces of a matrix Row-reduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,

### 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field

3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field 77 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field Overview: The antiderivative in one variable calculus is an important

### τ θ What is the proper price at time t =0of this option?

Now by Itô s formula But Mu f and u g in Ū. Hence τ θ u(x) =E( Mu(X) ds + u(x(τ θ))) 0 τ θ u(x) E( f(x) ds + g(x(τ θ))) = J x (θ). 0 But since u(x) =J x (θ ), we consequently have u(x) =J x (θ ) = min

### by the matrix A results in a vector which is a reflection of the given

Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

### NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that

NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called

### Numerical Methods for Differential Equations

Numerical Methods for Differential Equations Course objectives and preliminaries Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the Numerical Analysis

### Math 54. Selected Solutions for Week Is u in the plane in R 3 spanned by the columns

Math 5. Selected Solutions for Week 2 Section. (Page 2). Let u = and A = 5 2 6. Is u in the plane in R spanned by the columns of A? (See the figure omitted].) Why or why not? First of all, the plane in

### 1.5 Elementary Matrices and a Method for Finding the Inverse

.5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:

### Introduction to Green s Functions: Lecture notes 1

October 18, 26 Introduction to Green s Functions: Lecture notes 1 Edwin Langmann Mathematical Physics, KTH Physics, AlbaNova, SE-16 91 Stockholm, Sweden Abstract In the present notes I try to give a better

### Notes on wavefunctions

Notes on wavefunctions The double slit experiment In the double slit experiment, a beam of light is send through a pair of slits, and then observed on a screen behind the slits. At first, we might expect

### 3.4. Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes

Solving Simultaneous Linear Equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

### 4 Solving Systems of Equations by Reducing Matrices

Math 15 Sec S0601/S060 4 Solving Systems of Equations by Reducing Matrices 4.1 Introduction One of the main applications of matrix methods is the solution of systems of linear equations. Consider for example

### ECG590I Asset Pricing. Lecture 2: Present Value 1

ECG59I Asset Pricing. Lecture 2: Present Value 1 2 Present Value If you have to decide between receiving 1\$ now or 1\$ one year from now, then you would rather have your money now. If you have to decide

### Numerical Methods for Differential Equations

Numerical Methods for Differential Equations Chapter 1: Initial value problems in ODEs Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the Numerical

### State of Stress at Point

State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,

### EXAM. Practice Questions for Exam #2. Math 3350, Spring April 3, 2004 ANSWERS

EXAM Practice Questions for Exam #2 Math 3350, Spring 2004 April 3, 2004 ANSWERS i Problem 1. Find the general solution. A. D 3 (D 2)(D 3) 2 y = 0. The characteristic polynomial is λ 3 (λ 2)(λ 3) 2. Thus,

### PSTricks. pst-ode. A PSTricks package for solving initial value problems for sets of Ordinary Differential Equations (ODE), v0.7.

PSTricks pst-ode A PSTricks package for solving initial value problems for sets of Ordinary Differential Equations (ODE), v0.7 27th March 2014 Package author(s): Alexander Grahn Contents 2 Contents 1 Introduction

### Exponential Growth and Decay

Exponential Growth and Decay Recall that if y = f(t), then f (t) = dy dt y with respect to t. is called the rate of change of Another very important measure of rate of change is the relative rate of change

### 1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.

.(6pts Find symmetric equations of the line L passing through the point (, 5, and perpendicular to the plane x + 3y z = 9. (a x = y + 5 3 = z (b x (c (x = ( 5(y 3 = z + (d x (e (x + 3(y 3 (z = 9 = y 3

### AB2.2: Curves. Gradient of a Scalar Field

AB2.2: Curves. Gradient of a Scalar Field Parametric representation of a curve A a curve C in space can be represented by a vector function r(t) = [x(t), y(t), z(t)] = x(t)i + y(t)j + z(t)k This is called

### Chapter 2. Parameterized Curves in R 3

Chapter 2. Parameterized Curves in R 3 Def. A smooth curve in R 3 is a smooth map σ : (a, b) R 3. For each t (a, b), σ(t) R 3. As t increases from a to b, σ(t) traces out a curve in R 3. In terms of components,

### Notes from FE Review for mathematics, University of Kentucky

Notes from FE Review for mathematics, University of Kentucky Dr. Alan Demlow March 19, 2012 Introduction These notes are based on reviews for the mathematics portion of the Fundamentals of Engineering

### PHYS 1624 University Physics I. PHYS 2644 University Physics II

PHYS 1624 Physics I An introduction to mechanics, heat, and wave motion. This is a calculus- based course for Scientists and Engineers. 4 hours (3 lecture/3 lab) Prerequisites: Credit for MATH 2413 (Calculus

### HW 3 Due Sep 12, Wed

HW 3 Due Sep, Wed 1. Consider a tan used in certain hydrodynamic experiments. After one experiment the tan contains 200 L of a dye solution with a concentration of 1 g/l. To prepare for the next experiment,

### Integration and projectile motion (Sect. 13.2)

Interation and projectile motion (Sect. 13.) Interation of vector functions. Application: Projectile motion. Equations of a projectile motion. Rane, Heiht, Fliht Time. Interation of vector functions Definition

### Mechanics 1: Conservation of Energy and Momentum

Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation

### An optimal transportation problem with import/export taxes on the boundary

An optimal transportation problem with import/export taxes on the boundary Julián Toledo Workshop International sur les Mathématiques et l Environnement Essaouira, November 2012..................... Joint

### m = 4 m = 5 x x x t = 0 u = x 2 + 2t t < 0 t > 1 t = 0 u = x 3 + 6tx t > 1 t < 0

ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 21, Number 2, Spring 1991 NODAL PROPERTIES OF SOLUTIONS OF PARABOLIC EQUATIONS SIGURD ANGENENT * 1. Introduction. In this note we review the known facts about

### On principles of repulsive gravity: the Elementary Process Theory. M. Cabbolet (VUB) WAG2015

On principles of repulsive gravity: the Elementary Process Theory M. Cabbolet (VUB) WAG2015 overview acronyms: EPT: Elementary Process Theory GR: General Relativity WEP: Weak Equivalence Principle QM:

### 3.7 Non-autonomous linear systems of ODE. General theory

3.7 Non-autonomous linear systems of ODE. General theory Now I will study the ODE in the form ẋ = A(t)x + g(t), x(t) R k, A, g C(I), (3.1) where now the matrix A is time dependent and continuous on some

### The Method of Lagrange Multipliers

The Method of Lagrange Multipliers S. Sawyer October 25, 2002 1. Lagrange s Theorem. Suppose that we want to maximize (or imize a function of n variables f(x = f(x 1, x 2,..., x n for x = (x 1, x 2,...,