# tegrals as General & Particular Solutions

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 tegrals as General & Particular Solutions dy dx = f(x) General Solution: y(x) = f(x) dx + C Particular Solution: dy dx = f(x), y(x 0) = y 0 Examples: 1) dy dx = (x 2)2 ;y(2) = 1; 2) dy ;y(0) = 0; 3) dx = xe x ;y(0) = 1; dy dx = 10 x 2 +1 p. 2/3

2 tegrals as General and particular Solution Velocity and Acceleration A Swimmer s Problem (page 15): There is a northword-flowing river of width w = 2a with velocity ( ) v R = v 0 1 x2 a 2 for a x a. Suppose that a swimmer swims due east (relative to water) with constant speed v S. Find the swimmer s trajectory y = y(x) when he crosses the river. (Let s assume w = 1, v 0 = 9, and v S = 3) p. 3/3

3 Existence and Uniqueness of Solutions Theorem: Suppose that both function f(x,y) and its partial derivatives D y f(x,y) are continuous on some rectangle R in the xy-plane that contains the point (a,b) in its interior. Then, for some open interval I containing the point a, the initial value problem dy dx = f(x,y), y(a) = b has one and only one solution that is defined on the interval I Examples: 1) dy dy dx dx = x 1 dy,y(0) = 0; 2) dx = 2 y,y(0) = 0; 3) dy = y; 4) dx = y2,y(0) = 1 p. 2/8

4 Implicit Solutions and Singular Solutions Implicit Solutions: the equation K(x,y) = 0 is called an implicit solution of a differential equation if it is satisfies by some solution y = y(x) of the differential equation. Example: dy dx = 3y 4 2x 2 5 General Solutions Singular Solutions: Example: (y ) 2 = 4y Find all solutions of the differential equation xy y = 2x 2 y, y(1) = 1 p. 3/3

5 Linear First-Order Equations Linear first-order equation: dy dx + P(x)y = Q(x) The general solution of the linear first-order equation: y(x) = e [ ( ) ] P(x) dx Q(x)e P(x) dx dx + C Remark: We need not supply explicitly a constant of integration when we find the integrating factor ρ(x). p. 2/4

6 Linear First-Order Equations Theorem: If the functions P(x) and Q(x) are continuous on the open interval I containing the point x 0, then the initial value equation dy dx + P(x)y = Q(x), y(x 0) = y 0 has a unique solution y(x) on I, given by the formula y(x) = e [ ( ) ] P(x) dx Q(x)e P(x) dx dx + C with an appropriate value of C. p. 4/4

7 Chapter 1: First-Order Differential Equations Section 1.6: Substitution Methods and Exact Equations p. 1/7

8 First-Order Equations dy dx = F(ax + by + c) Step 1: Let v(x) = ax + by + c Step 2: dv dx = a + bdy dx Step 3: dx dv = a + bf(v). This is a separable first-order differential equation Example: y = x + y + 1 p. 2/7

9 Homogeneous First-Order Equations dy ( y ) dx = F x Step 1: Let v(x) = y x Step 2: dy dx = v + xdv dx Step 3: xdx dv = F(v) v. This is a separable first-order differential equation Example: x 2 y = xy + x 2 e y/x Example: yy + x = x 2 + y 2 p. 3/7

10 Bernoulli Equations dy dx + P(x)y = Q(x)yn Step 0: If n = 0, it is a linear equation; If n = 1, it is a separable equation. Step 1: Let v(x) = y 1 n for n 0, 1 Step 2: dv dx = (1 n)y n dy dx Step 3: dx dv + (1 n)p(x)v = (1 n)q(x). This is a linear first-order differential equation Example: y = y + y 3 Example: x dy dx + 6y = 3xy4/3 p. 4/7

11 Definition A(x)y + B(x)y + C(x)y = F(x) Homogeneous linear equation: F(x) = 0 Non-Homogeneous linear equation: F(x) 0 p. 2/8

12 Theorems Theorem: Let y 1 and y 2 be two solutions of the homogeneous linear equation y + p(x)y + q(x)y = 0 on the interval I. If c 1 and c 2 are constants, then the linear combination y = c 1 y 1 + c 2 y 2 is also a solution of above equation on I p. 3/8

13 Theorems Theorem: Suppose that the functions p, q, and f are continuous on the open interval I containing the point a. Then, given any two numbers b 0 and b 1, the equation y + p(x)y + q(x)y = f(x) has a unique solution on the entire interval I that satisfies the initial conditions y(a) = b 0, y (a) = b 1 p. 4/8

14 Linearly Independent Solutions Definition: Two functions defined on an open interval I are said to be linearly independent on I provided that neither is a constant multiple of the other. Wronskian: The Wronskian of f and g is the determinant f g W(f,g) = f g p. 5/8

15 Theorems Theorem: Suppose that y 1 and y 2 are two solutions of the homogeneous second-order linear equation y + p(x)y + q(x)y = 0 on an open interval I on which p and q are continuous. Then 1. If y 1 and y 2 are linearly dependent, then W(y 1,y 2 ) 0 on I 2. If y 1 and y 2 are linearly independent, then W(y 1,y 2 ) 0 at each point of I p. 6/8

16 Theorems Theorem: Let y 1 and y 2 be two linear independent solutions of the homogeneous second-order linear equation y + p(x)y + q(x)y = 0 on an open interval I on which p and q are continuous. If Y is any solution whatsoever of above equation on I, then there exist numbers c 1 and c 2 such that for all x in I. Y (x) = c 1 y 1 (x) + c 2 y 2 (x) p. 7/8

17 Theorems The quadratic equation ar 2 + br + c = 0 is called the characteristic equation of the homogeneous second-order linear differential equation ay + by + cy = 0. Theorem: Let r 1 and r 2 be real and distinct roots of the characteristic equation, then y(x) = c 1 e r 1x + c 2 e r 2x is the general solution of the homogeneous second-order linear differential equation. p. 8/8

18 Theorems Theorem: Let r 1 be repeated root of the characteristic equation, then y(x) = (c 1 + c 2 x)e r 1x is the general solution of the homogeneous second-order linear differential equation. p. 9/8

19 Characteristic Equation a n y (n) + a n 1 y (n 1) + a 1 y + a 0 y = 0 with a n 0. The characteristic function of above differential equation is a n r n + a n 1 r n 1 + a 1 r + a 0 = 0 p. 2/7

20 Theorem Theorem: If the roots r 1,r 2,...,r n of the characteristic function of the differential equation with constant coefficients are real and distinct, then y(x) = c 1 e r 1x + c 2 e r 2x + + c n e r nx is a general solution of the given equation. Thus the n linearly independent functions {e r 1x,e r 2x,...,e r nx } constitute a basis for the n-dimensional solution space. Example: Solve the initial value problem y (3) + 3y 10y = 0 with y(0) = 7,y (0) = 0,y (0) = 70. p. 3/7

21 Theorem Theorem: If the characteristic function of the differential equation with constant coefficients has a repeated root r of multiplicity k, then the part of a general solution of the differential equation corresponding to r is of the form (c 1 + c 2 x + c 3 x c k x k 1 )e rx Example: 9y (5) 6y (4) + y (3) = 0 Example: y (4) 3 (3) + 3y y = 0 Example: y (4) 8y + 16y = 0 Example: y (3) + y y y = 0 p. 5/7

22 plex-valued Functions and Euler s Form Euler s formula: Some Facts: e ix = cosx + i sin x e (a+ib)x = e ax (cosbx + i sin bx) e (a ib)x = e ax (cosbx i sin bx) D x (e rx ) = re rx p. 6/7

23 Theorem Theorem: If the characteristic function of the differential equation with constant coefficients has an unrepeated pair of complex conjugate roots a ± bi (with b 0), then the correspondence part of a general solution of the equation has the form e ax (c 1 cosbx + c 2 sin bx) Thus the linearly independent solutions e ax cos bx and e ax sin bx generate a 2-dimensional subspace of the solution space of the differential equation. Example: y (4) + 4y = 0 Example: y (4) + 3y 4y = 0 Example: y (4) = 16y p. 7/7

24

25

26

27

28

29

30 Chapter 9 Ordinary differential equations Solve y + ky = f(x) for f(x) a piecewise continuous function with period 2L. Heat conduction u t = ku xx, u(0, t) = u(l, t) = 0, u(x, 0) = f(x), 0 < x < L, t > 0; u t = ku xx, u x (0, t) = u x (L, t) = 0, u(x, 0) = f(x), 0 < x < L, t > 0; (1a) (1b) (1c) (2a) (2b) (2c) (1a)-(1c) is listed on textbook page 608; Solution is given by Theorem 1 on page 613 u(x, t) = n=1 b n exp( n2 π 2 kt L 2 ) sin nπx L.

31 with b n the fourier sine coefficients of f(x). (2a)-(2c) is listed on textbook page 615, the solution is given as Theorem 2 on page 616. y(x, t) = a with a n f(x). n=1 a n exp( n2 π 2 kt L 2 ) cos nπx L. the fourier cosine coefficients of Vibrating strings y tt = a 2 y xx, y(0, t) = y(l, t) = 0, y(x, 0) = f(x), 0 < x < L, t > 0 y t (x, 0) = 0 (3a) (3b) (3c) (3d) y tt = a 2 y xx, y(0, t) = y(l, t) = 0, y(x, 0) = 0, 0 < x < L, t > 0 y t (x, 0) = g(x). (4a) (4b) (4c) (4d)

32 (3a)-(3d) is listed as Problem A on page 623. The solution is given as (22) and (23) on page 625 of the textbook. y(x, t) = n=1 b n cos nπa nπx t sin L L with b n the fourier sine coefficient of f(x). (4a)-(4d) is listed as Problem B on page 623. The solution is given as (33) and (35) on page 629 of the textbook. y(x, t) = n=1 b n nπa sin nπat L nπx sin L with b n the fourier sine coefficient of g(x).

33 Uxx + Uyy = 0 U(o,y) = U(a,y) = U(x,b) = 0 U(x,o) = f(x) Uxx + Uyy = 0 U(o,y) = U(a,y) = U(x,o) = 0 U(x,b) = f(x) Uxx + Uyy = 0 U(x,b) = U(a,y) = U(x,o) = 0 U(o,y) = g(x) Uxx + Uyy = 0 U(x,b) = U(o,y) = U(x,o) = 0 U(a,y) = g(x)

### Methods of Solution of Selected Differential Equations Carol A. Edwards Chandler-Gilbert Community College

Methods of Solution of Selected Differential Equations Carol A. Edwards Chandler-Gilbert Community College Equations of Order One: Mdx + Ndy = 0 1. Separate variables. 2. M, N homogeneous of same degree:

### Second-Order Linear Differential Equations

Second-Order Linear Differential Equations A second-order linear differential equation has the form 1 Px d 2 y dx 2 dy Qx dx Rxy Gx where P, Q, R, and G are continuous functions. We saw in Section 7.1

### A Brief Review of Elementary Ordinary Differential Equations

1 A Brief Review of Elementary Ordinary Differential Equations At various points in the material we will be covering, we will need to recall and use material normally covered in an elementary course on

### 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

### SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS

SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS A second-order linear differential equation has the form 1 Px d y dx dy Qx dx Rxy Gx where P, Q, R, and G are continuous functions. Equations of this type arise

### 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

### 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

### College of the Holy Cross, Spring 2009 Math 373, Partial Differential Equations Midterm 1 Practice Questions

College of the Holy Cross, Spring 29 Math 373, Partial Differential Equations Midterm 1 Practice Questions 1. (a) Find a solution of u x + u y + u = xy. Hint: Try a polynomial of degree 2. Solution. Use

### 19.6. Finding a Particular Integral. Introduction. Prerequisites. Learning Outcomes. Learning Style

Finding a Particular Integral 19.6 Introduction We stated in Block 19.5 that the general solution of an inhomogeneous equation is the sum of the complementary function and a particular integral. We have

### Nonhomogeneous Linear Equations

Nonhomogeneous Linear Equations In this section we learn how to solve second-order nonhomogeneous linear differential equations with constant coefficients, that is, equations of the form ay by cy G x where

### 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

### Ordinary Differential Equations

Ordinary Differential Equations Dan B. Marghitu and S.C. Sinha 1 Introduction An ordinary differential equation is a relation involving one or several derivatives of a function y(x) with respect to x.

### The Dirichlet Problem on a Rectangle

The Dirichlet Problem on a Rectangle R. C. Trinity University Partial Differential Equations March 18, 014 The D heat equation Steady state solutions to the -D heat equation Laplace s equation Recall:

### Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if

1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c

### Math 432 HW 2.5 Solutions

Math 432 HW 2.5 Solutions Assigned: 1-10, 12, 13, and 14. Selected for Grading: 1 (for five points), 6 (also for five), 9, 12 Solutions: 1. (2y 3 + 2y 2 ) dx + (3y 2 x + 2xy) dy = 0. M/ y = 6y 2 + 4y N/

### Nonconstant Coefficients

Chapter 7 Nonconstant Coefficients We return to second-order linear ODEs, but with nonconstant coefficients. That is, we consider 7.1 y +pty +qty = 0, with not both pt and qt constant. The theory developed

### 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

### f(x) = lim 2) = 2 2 = 0 (c) Provide a rough sketch of f(x). Be sure to include your scale, intercepts and label your axis.

Math 16 - Final Exam Solutions - Fall 211 - Jaimos F Skriletz 1 Answer each of the following questions to the best of your ability. To receive full credit, answers must be supported by a sufficient amount

### 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,

### Introduction to polynomials

Worksheet 4.5 Polynomials Section 1 Introduction to polynomials A polynomial is an expression of the form p(x) = p 0 + p 1 x + p 2 x 2 + + p n x n, (n N) where p 0, p 1,..., p n are constants and x os

### Separable First Order Differential Equations

Separable First Order Differential Equations Form of Separable Equations which take the form = gx hy or These are differential equations = gxĥy, where gx is a continuous function of x and hy is a continuously

### Math 201 Lecture 23: Power Series Method for Equations with Polynomial

Math 201 Lecture 23: Power Series Method for Equations with Polynomial Coefficients Mar. 07, 2012 Many examples here are taken from the textbook. The first number in () refers to the problem number in

### CHAPTER 2 FOURIER SERIES

CHAPTER 2 FOURIER SERIES PERIODIC FUNCTIONS A function is said to have a period T if for all x,, where T is a positive constant. The least value of T>0 is called the period of. EXAMPLES We know that =

### INTERPOLATION. Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y).

INTERPOLATION Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y). As an example, consider defining and x 0 =0, x 1 = π 4, x

### 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)

### On closed-form solutions to a class of ordinary differential equations

International Journal of Advanced Mathematical Sciences, 2 (1 (2014 57-70 c Science Publishing Corporation www.sciencepubco.com/index.php/ijams doi: 10.14419/ijams.v2i1.1556 Research Paper On closed-form

### The two dimensional heat equation

The two dimensional heat equation Ryan C. Trinity University Partial Differential Equations March 6, 2012 Physical motivation Consider a thin rectangular plate made of some thermally conductive material.

### 1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style

Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with

### 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

### 4 Linear Dierential Equations

Dierential Equations (part 2): Linear Dierential Equations (by Evan Dummit, 2012, v. 1.00) Contents 4 Linear Dierential Equations 1 4.1 Terminology.................................................. 1 4.2

### a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

### 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

### 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

### Elementary Differential Equations

Elementary Differential Equations EIGHTH EDITION Earl D. Rainville Late Professor of Mathematics University of Michigan Phillip E. Bedient Professor Emeritus of Mathematics Franklin and Marshall College

### 2 Integrating Both Sides

2 Integrating Both Sides So far, the only general method we have for solving differential equations involves equations of the form y = f(x), where f(x) is any function of x. The solution to such an equation

### Functions and Equations

Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### CALCULUS 2. 0 Repetition. tutorials 2015/ Find limits of the following sequences or prove that they are divergent.

CALCULUS tutorials 5/6 Repetition. Find limits of the following sequences or prove that they are divergent. a n = n( ) n, a n = n 3 7 n 5 n +, a n = ( n n 4n + 7 ), a n = n3 5n + 3 4n 7 3n, 3 ( ) 3n 6n

### Module 3: Second-Order Partial Differential Equations

Module 3: Second-Order Partial Differential Equations In Module 3, we shall discuss some general concepts associated with second-order linear PDEs. These types of PDEs arise in connection with various

### 5 Indefinite integral

5 Indefinite integral The most of the mathematical operations have inverse operations: the inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse

Solving Second Order Linear ODEs Table of contents Solving Second Order Linear ODEs........................................ 1 1. Introduction: A few facts............................................. 1.

### Homework #1 Solutions

MAT 303 Spring 203 Homework # Solutions Problems Section.:, 4, 6, 34, 40 Section.2:, 4, 8, 30, 42 Section.4:, 2, 3, 4, 8, 22, 24, 46... Verify that y = x 3 + 7 is a solution to y = 3x 2. Solution: From

### Worksheet 4.7. Polynomials. Section 1. Introduction to Polynomials. A polynomial is an expression of the form

Worksheet 4.7 Polynomials Section 1 Introduction to Polynomials A polynomial is an expression of the form p(x) = p 0 + p 1 x + p 2 x 2 + + p n x n (n N) where p 0, p 1,..., p n are constants and x is a

### 6.1 Add & Subtract Polynomial Expression & Functions

6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic

### Finding Antiderivatives and Evaluating Integrals

Chapter 5 Finding Antiderivatives and Evaluating Integrals 5. Constructing Accurate Graphs of Antiderivatives Motivating Questions In this section, we strive to understand the ideas generated by the following

### Introduction to Partial Differential Equations. John Douglas Moore

Introduction to Partial Differential Equations John Douglas Moore May 2, 2003 Preface Partial differential equations are often used to construct models of the most basic theories underlying physics and

### Math Spring Review Material - Exam 3

Math 85 - Spring 01 - Review Material - Exam 3 Section 9. - Fourier Series and Convergence State the definition of a Piecewise Continuous function. Answer: f is Piecewise Continuous if the following to

### Finding y p in Constant-Coefficient Nonhomogenous Linear DEs

Finding y p in Constant-Coefficient Nonhomogenous Linear DEs Introduction and procedure When solving DEs of the form the solution looks like ay + by + cy =, y = y c + y p where y c is the complementary

### HOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba

HOMEWORK SOLUTIONS All questions are from Vector Calculus, by Marsden and Tromba Question :..6 Let w = f(x, y) be a function of two variables, and let x = u + v, y = u v. Show that Solution. By the chain

### INTEGRATING FACTOR METHOD

Differential Equations INTEGRATING FACTOR METHOD Graham S McDonald A Tutorial Module for learning to solve 1st order linear differential equations Table of contents Begin Tutorial c 2004 g.s.mcdonald@salford.ac.uk

### College Algebra - MAT 161 Page: 1 Copyright 2009 Killoran

College Algebra - MAT 6 Page: Copyright 2009 Killoran Zeros and Roots of Polynomial Functions Finding a Root (zero or x-intercept) of a polynomial is identical to the process of factoring a polynomial.

### MATH 461: Fourier Series and Boundary Value Problems

MATH 461: Fourier Series and Boundary Value Problems Chapter III: Fourier Series Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2015 fasshauer@iit.edu MATH 461 Chapter

### x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1

Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs

### Introduction to Algebraic Geometry. Bézout s Theorem and Inflection Points

Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a

### Differential Equations BERNOULLI EQUATIONS. Graham S McDonald. A Tutorial Module for learning how to solve Bernoulli differential equations

Differential Equations BERNOULLI EQUATIONS Graham S McDonald A Tutorial Module for learning how to solve Bernoulli differential equations Table of contents Begin Tutorial c 2004 g.s.mcdonald@salford.ac.uk

### , < x < Using separation of variables, u(x, t) = Φ(x)h(t) (4) we obtain the differential equations. d 2 Φ = λφ (6) Φ(± ) < (7)

Chapter1: Fourier Transform Solutions of PDEs In this chapter we show how the method of separation of variables may be extended to solve PDEs defined on an infinite or semi-infinite spatial domain. Several

### 1 Introduction to Differential Equations

1 Introduction to Differential Equations A differential equation is an equation that involves the derivative of some unknown function. For example, consider the equation f (x) = 4x 3. (1) This equation

### QUADRATIC SYSTEMS WITH A RATIONAL FIRST INTEGRAL OF DEGREE THREE: A COMPLETE CLASSIFICATION IN THE COEFFICIENT SPACE R 12

QUADRATIC SYSTEMS WITH A RATIONAL FIRST INTEGRAL OF DEGREE THREE: A COMPLETE CLASSIFICATION IN THE COEFFICIENT SPACE R 12 JOAN C. ARTÉS1, JAUME LLIBRE 1 AND NICOLAE VULPE 2 Abstract. A quadratic polynomial

### Notes and questions to aid A-level Mathematics revision

Notes and questions to aid A-level Mathematics revision Robert Bowles University College London October 4, 5 Introduction Introduction There are some students who find the first year s study at UCL and

### Differentiation and Integration

This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have

### Solutions for Review Problems

olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector

### Chapter 4. Linear Second Order Equations. ay + by + cy = 0, (1) where a, b, c are constants. The associated auxiliary equation is., r 2 = b b 2 4ac 2a

Chapter 4. Linear Second Order Equations ay + by + cy = 0, (1) where a, b, c are constants. ar 2 + br + c = 0. (2) Consequently, y = e rx is a solution to (1) if an only if r satisfies (2). So, the equation

### SOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve

SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives

Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the

### 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

### Fourier Series and Sturm-Liouville Eigenvalue Problems

Fourier Series and Sturm-Liouville Eigenvalue Problems 2009 Outline Functions Fourier Series Representation Half-range Expansion Convergence of Fourier Series Parseval s Theorem and Mean Square Error Complex

### MATH 381 HOMEWORK 2 SOLUTIONS

MATH 38 HOMEWORK SOLUTIONS Question (p.86 #8). If g(x)[e y e y ] is harmonic, g() =,g () =, find g(x). Let f(x, y) = g(x)[e y e y ].Then Since f(x, y) is harmonic, f + f = and we require x y f x = g (x)[e

### 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

### NUMERICAL ANALYSIS PROGRAMS

NUMERICAL ANALYSIS PROGRAMS I. About the Program Disk This disk included with Numerical Analysis, Seventh Edition by Burden and Faires contains a C, FORTRAN, Maple, Mathematica, MATLAB, and Pascal program

### CHAPTER 4. Test Bank Exercises in. Exercise Set 4.1

Test Bank Exercises in CHAPTER 4 Exercise Set 4.1 1. Graph the quadratic function f(x) = x 2 2x 3. Indicate the vertex, axis of symmetry, minimum 2. Graph the quadratic function f(x) = x 2 2x. Indicate

### x + 5 x 2 + x 2 dx Answer: ln x ln x 1 + c

. Evaluate the given integral (a) 3xe x2 dx 3 2 e x2 + c (b) 3 x ln xdx 2x 3/2 ln x 4 3 x3/2 + c (c) x + 5 x 2 + x 2 dx ln x + 2 + 2 ln x + c (d) x sin (πx) dx x π cos (πx) + sin (πx) + c π2 (e) 3x ( +

### 6.4 The Remainder Theorem

6.4. THE REMAINDER THEOREM 6.3.2 Check the two divisions we performed in Problem 6.12 by multiplying the quotient by the divisor, then adding the remainder. 6.3.3 Find the quotient and remainder when x

### Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.

Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method

### AB2.14: Heat Equation: Solution by Fourier Series

AB2.14: Heat Equation: Solution by Fourier Series Consider the boundary value problem for the one-dimensional heat equation describing the temperature variation in a bar with the zero-temperature ends:

### Fourier Series Chapter 3 of Coleman

Fourier Series Chapter 3 of Coleman Dr. Doreen De eon Math 18, Spring 14 1 Introduction Section 3.1 of Coleman The Fourier series takes its name from Joseph Fourier (1768-183), who made important contributions

### THE DIFFUSION EQUATION

THE DIFFUSION EQUATION R. E. SHOWALTER 1. Heat Conduction in an Interval We shall describe the diffusion of heat energy through a long thin rod G with uniform cross section S. As before, we identify G

### Limits and Continuity

Math 20C Multivariable Calculus Lecture Limits and Continuity Slide Review of Limit. Side limits and squeeze theorem. Continuous functions of 2,3 variables. Review: Limits Slide 2 Definition Given a function

### Recognizing Types of First Order Differential Equations E. L. Lady

Recognizing Types of First Order Differential Equations E. L. Lady Every first order differential equation to be considered here can be written can be written in the form P (x, y)+q(x, y)y =0. This means

### 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

### Recitation 4. 24xy for 0 < x < 1, 0 < y < 1, x + y < 1 0 elsewhere

Recitation. Exercise 3.5: If the joint probability density of X and Y is given by xy for < x

### Second Order Linear Differential Equations

CHAPTER 2 Second Order Linear Differential Equations 2.. Homogeneous Equations A differential equation is a relation involving variables x y y y. A solution is a function f x such that the substitution

### Zeros of Polynomial Functions

Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

Week 5 Vladimir Dobrushkin 5. Fourier Series In this section, we present a remarkable result that gives the matehmatical explanation for how cellular phones work: Fourier series. It was discovered in the

### 2 First-Order Equations: Method of Characteristics

2 First-Order Equations: Method of Characteristics In this section, we describe a general technique for solving first-order equations. We begin with linear equations and work our way through the semilinear,

### Chapter 14: Fourier Transforms and Boundary Value Problems in an Unbounded Region

Chapter 14: Fourier Transforms and Boundary Value Problems in an Unbounded Region 王奕翔 Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw December 25, 213 1 / 27 王奕翔 DE Lecture

### correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:

Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that

### Fourier series. Jan Philip Solovej. English summary of notes for Analysis 1. May 8, 2012

Fourier series Jan Philip Solovej English summary of notes for Analysis 1 May 8, 2012 1 JPS, Fourier series 2 Contents 1 Introduction 2 2 Fourier series 3 2.1 Periodic functions, trigonometric polynomials

### MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

### COMPLEX NUMBERS. a bi c di a c b d i. a bi c di a c b d i For instance, 1 i 4 7i 1 4 1 7 i 5 6i

COMPLEX NUMBERS _4+i _-i FIGURE Complex numbers as points in the Arg plane i _i +i -i A complex number can be represented by an expression of the form a bi, where a b are real numbers i is a symbol with

### 1 Lecture 19: Implicit differentiation

Lecture 9: Implicit differentiation. Outline The technique of implicit differentiation Tangent lines to a circle Examples.2 Implicit differentiation Suppose we have two quantities or variables x and y

### Solving Cubic Polynomials

Solving Cubic Polynomials 1.1 The general solution to the quadratic equation There are four steps to finding the zeroes of a quadratic polynomial. 1. First divide by the leading term, making the polynomial

### Sine and Cosine Series; Odd and Even Functions

Sine and Cosine Series; Odd and Even Functions A sine series on the interval [, ] is a trigonometric series of the form k = 1 b k sin πkx. All of the terms in a series of this type have values vanishing

### This makes sense. t 2 1 + 1/t 2 dt = 1. t t 2 + 1dt = 2 du = 1 3 u3/2 u=5

1. (Line integrals Using parametrization. Two types and the flux integral) Formulas: ds = x (t) dt, d x = x (t)dt and d x = T ds since T = x (t)/ x (t). Another one is Nds = T ds ẑ = (dx, dy) ẑ = (dy,

### MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.

MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α

### Math 20D Midterm Exam Practice Problems Solutions

Math 20D Midterm Exam Practice Problems Solutions 1. A tank contains G gallons of fresh water. A solution with a concentration of m lb of salt per gallon is pumped into the tank at a rate of r gallons

### Fourier Series. A Fourier series is an infinite series of the form. a + b n cos(nωx) +

Fourier Series A Fourier series is an infinite series of the form a b n cos(nωx) c n sin(nωx). Virtually any periodic function that arises in applications can be represented as the sum of a Fourier series.

### 19.7. Applications of Differential Equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Applications of Differential Equations 19.7 Introduction Blocks 19.2 to 19.6 have introduced several techniques for solving commonly-occurring firstorder and second-order ordinary differential equations.

7 Quadratic Expressions A quadratic expression (Latin quadratus squared ) is an expression involving a squared term, e.g., x + 1, or a product term, e.g., xy x + 1. (A linear expression such as x + 1 is