mathcentre community project
|
|
- Rosaline Nichols
- 6 years ago
- Views:
Transcription
1 community project mathcentre community project First Order Ordinary All mccp resourcesdifferential are released under a Creative Equation Commons licence Introduction mccp-richard- Prerequisites: You will need to know about trigonometry, dierentiation, integration, complex numbers in order to make the most of this teach-yourself resource. We are looking at equations involving a function y(x) and its rst derivative: + P(x)y = Q(x) () We want to nd y(x), either explicitly if possible, or otherwise implicitly. Direct Integration These ODEs can be solved as follows: = f(x) = f(x) so: y = f(x) = 3x2 6x + 5 y = x 3 3x 2 + 5x + C with C constant of integration The constant of integration can be anything, unless you have boundary conditions. In the previous example, if you have y(0)=0 then: y(0) = C = 0
2 Separation of Variables = f(x)g(y) mathcentre community project These ODEs can be solved as follows: All mccp resources are released under a Creative Commons licence = f(x) g(y) = f(x) g(y) = 2x y + (y + ) = 2x (y + ) = 2x 2 y2 + y = x 2 + C with C constant of integration Homogeneous Equations A rst order homogeneous dierential equation is a dierential equation in the form: = f(y x ) These ODEs can be solved by making the substitution y(x) = v(x) x where v is a function of x. Then we have: Using the product rule: Inserting in the ODE: Re-arranging: = x + v x + v = f(v) f(v) v = x f(v) v = x
3 Example Rearranging, we get: = x + 3y 2x y mathcentre = 2 community x x project + v = v All mccp resources x are released under a Creative Commons licence = v + v = 2x Taking the log on both sides: v = e C x /2 with y = v x, ln( + v) = 2 ln(x) + C = ln(x/2 ) + C withc constant of integration y = kx 3/2 x, with k = e C Integrating Factor + P(x)y = Q(x) These ODEs can be solved as follows: multiply both sides of the equation by the integrating factor: e P(x). Then: e P(x) + P(x)e P(x) y = Q(x)e P(x) Now we see that, using the product rule and the chain rule: e P(x) + P(x)e P(x) y = d ( e ) P(x) y Therefore: d ( e ) P(x) y = Q(x)e P(x) y = e P(x) Q(x)e P(x) Example The integrating factor is e = e x and: y = x with P(x) = and Q(x) = x x e e x y = xe x d ( ) ye x = xe x Using integration by part: ye x = e x ( + x) + C with C constant of integration y = ( + x) + Ce x
4 Bernouilli Equations Bernouilli equations are of the form: + P(x)y = Q(x)yn mathcentre These equations are solved with the following community method: project All mccp resources are released under a Creative Commons licence Multiply both sides of the equation by y n, the equation becomes: Make the change of variable v = y n, then: n y + P(x)y n = Q(x) so + P(x)v = Q(x) n = ( n)y n The latest form of the equation can be solved for v using the method of the integrating factor. Finally, y can be found from the relationship y = v n. y 2 + x y = xy2 + x y = x v = y, v x = x = y 2 The integrating factor is IF = e x = e ln x = x x x v = 2 x v = v = x 2 + Cx y = x 2 + Cx
5 Exercises (a) x = 5x3 + 4 (b) x(y 3) (d) x y = x3 + 3x 2 2x (e) ( + x2 ) mathcentre = 4y + 3xy = 5x with y() = 2 (c) (2y x) community project = 2x + y with y(2) = 3 (f) 2 Answers encouraging academics to share + y = y3 (x ) maths support resources All mccp resources are released under a Creative Commons licence (a) y = 5 3 x3 + 4 ln x + C (d) y = 2 x3 + 3x 2 2x ln x + Cx (b) y 3 ln y = 4 ln x + C (e) y = ( + x2 ) 3/2 (c) ( y x )2 y x + x 2 = 0 (f) y = (x + Ce x ) /2
JUST THE MATHS UNIT NUMBER ORDINARY DIFFERENTIAL EQUATIONS 3 (First order equations (C)) A.J.Hobson
JUST THE MATHS UNIT NUMBER 15.3 ORDINARY DIFFERENTIAL EQUATIONS 3 (First order equations (C)) by A.J.Hobson 15.3.1 Linear equations 15.3.2 Bernouilli s equation 15.3.3 Exercises 15.3.4 Answers to exercises
More informationElementary ODE Review
Elementary ODE Review First Order ODEs First Order Equations Ordinary differential equations of the fm y F(x, y) () are called first der dinary differential equations. There are a variety of techniques
More informationFirst Order Differential Equations Chapter 1
First Order Differential Equations Chapter 1 Doreen De Leon Department of Mathematics, California State University, Fresno 1 Differential Equations and Mathematical Models Section 1.1 Definitions: An equation
More informationPower Series and Analytic Function
Dr Mansoor Alshehri King Saud University MATH204-Differential Equations Center of Excellence in Learning and Teaching 1 / 21 Some Reviews of Power Series Differentiation and Integration of a Power Series
More informationOrdinary Differential Equations
Ordinary Differential Equations (MA102 Mathematics II) Shyamashree Upadhyay IIT Guwahati Shyamashree Upadhyay ( IIT Guwahati ) Ordinary Differential Equations 1 / 25 First order ODE s We will now discuss
More informationMA Ordinary Differential Equations
MA 108 - Ordinary Differential Equations Santanu Dey Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 76 dey@math.iitb.ac.in March 26, 2014 Outline of the lecture Method
More information4 Differential Equations
Advanced Calculus Chapter 4 Differential Equations 65 4 Differential Equations 4.1 Terminology Let U R n, and let y : U R. A differential equation in y is an equation involving y and its (partial) derivatives.
More information4 Exact Equations. F x + F. dy dx = 0
Chapter 1: First Order Differential Equations 4 Exact Equations Discussion: The general solution to a first order equation has 1 arbitrary constant. If we solve for that constant, we can write the general
More informationEXAM. Exam #1. Math 3350 Summer II, July 21, 2000 ANSWERS
EXAM Exam #1 Math 3350 Summer II, 2000 July 21, 2000 ANSWERS i 100 pts. Problem 1. 1. In each part, find the general solution of the differential equation. dx = x2 e y We use the following sequence of
More informationMore Techniques. for Solving First Order ODE'S. and. a Classification Scheme for Techniques
A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 1 A COLLECTION OF HANDOUTS ON FIRST ORDER ORDINARY DIFFERENTIAL
More informationFirst Order Differential Equations
Chapter 2 First Order Differential Equations Introduction Any first order differential equation can be written as F (x, y, y )=0 by moving all nonzero terms to the left hand side of the equation. Of course,
More information6 Second Order Linear Differential Equations
6 Second Order Linear Differential Equations A differential equation for an unknown function y = f(x) that depends on a variable x is any equation that ties together functions of x with y and its derivatives.
More informationDiff. Eq. App.( ) Midterm 1 Solutions
Diff. Eq. App.(110.302) Midterm 1 Solutions Johns Hopkins University February 28, 2011 Problem 1.[3 15 = 45 points] Solve the following differential equations. (Hint: Identify the types of the equations
More informationy x 3. Solve each of the given initial value problems. (a) y 0? xy = x, y(0) = We multiply the equation by e?x, and obtain Integrating both sides with
Solutions to the Practice Problems Math 80 Febuary, 004. For each of the following dierential equations, decide whether the given function is a solution. (a) y 0 = (x + )(y? ), y =? +exp(x +x)?exp(x +x)
More informationSeries Solutions Near a Regular Singular Point
Series Solutions Near a Regular Singular Point MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Background We will find a power series solution to the equation:
More informationJUST THE MATHS UNIT NUMBER ORDINARY DIFFERENTIAL EQUATIONS 1 (First order equations (A)) A.J.Hobson
JUST THE MATHS UNIT NUMBER 5. ORDINARY DIFFERENTIAL EQUATIONS (First order equations (A)) by A.J.Hobson 5.. Introduction and definitions 5..2 Exact equations 5..3 The method of separation of the variables
More informationSeries Solutions of Differential Equations
Chapter 6 Series Solutions of Differential Equations In this chapter we consider methods for solving differential equations using power series. Sequences and infinite series are also involved in this treatment.
More informationMath 240 Calculus III
Calculus III Summer 2015, Session II Monday, August 3, 2015 Agenda 1. 2. Introduction The reduction of technique, which applies to second- linear differential equations, allows us to go beyond equations
More informationA Brief Review of Elementary Ordinary Differential Equations
A 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
More informationTitle: Solving Ordinary Differential Equations (ODE s)
... Mathematics Support Centre Title: Solving Ordinary Differential Equations (ODE s) Target: On completion of this workbook you should be able to recognise and apply the appropriate method for solving
More information2. Second-order Linear Ordinary Differential Equations
Advanced Engineering Mathematics 2. Second-order Linear ODEs 1 2. Second-order Linear Ordinary Differential Equations 2.1 Homogeneous linear ODEs 2.2 Homogeneous linear ODEs with constant coefficients
More informationMath 201 Solutions to Assignment 1. 2ydy = x 2 dx. y = C 1 3 x3
Math 201 Solutions to Assignment 1 1. Solve the initial value problem: x 2 dx + 2y = 0, y(0) = 2. x 2 dx + 2y = 0, y(0) = 2 2y = x 2 dx y 2 = 1 3 x3 + C y = C 1 3 x3 Notice that y is not defined for some
More information1.11 Some Higher-Order Differential Equations
page 99. Some Higher-Order Differential Equations 99. Some Higher-Order Differential Equations So far we have developed analytical techniques only for solving special types of firstorder differential equations.
More information2.2 Separable Equations
2.2 Separable Equations Definition A first-order differential equation that can be written in the form Is said to be separable. Note: the variables of a separable equation can be written as Examples Solve
More informationEssential Ordinary Differential Equations
MODULE 1: MATHEMATICAL PRELIMINARIES 10 Lecture 2 Essential Ordinary Differential Equations In this lecture, we recall some methods of solving first-order IVP in ODE (separable and linear) and homogeneous
More informationOrdinary Differential Equations
Orinary Differential Equations Example: Harmonic Oscillator For a perfect Hooke s-law spring,force as a function of isplacement is F = kx Combine with Newton s Secon Law: F = ma with v = a = v = 2 x 2
More informationSection Summary. Section 1.5 9/9/2014
Section 1.5 Section Summary Nested Quantifiers Order of Quantifiers Translating from Nested Quantifiers into English Translating Mathematical Statements into Statements involving Nested Quantifiers Translated
More informationECONOMICS 207 SPRING 2006 LABORATORY EXERCISE 5 KEY. 8 = 10(5x 2) = 9(3x + 8), x 50x 20 = 27x x = 92 x = 4. 8x 2 22x + 15 = 0 (2x 3)(4x 5) = 0
ECONOMICS 07 SPRING 006 LABORATORY EXERCISE 5 KEY Problem. Solve the following equations for x. a 5x 3x + 8 = 9 0 5x 3x + 8 9 8 = 0(5x ) = 9(3x + 8), x 0 3 50x 0 = 7x + 7 3x = 9 x = 4 b 8x x + 5 = 0 8x
More information17.2 Nonhomogeneous Linear Equations. 27 September 2007
17.2 Nonhomogeneous Linear Equations 27 September 2007 Nonhomogeneous Linear Equations The differential equation to be studied is of the form ay (x) + by (x) + cy(x) = G(x) (1) where a 0, b, c are given
More informationLesson 3: Linear differential equations of the first order Solve each of the following differential equations by two methods.
Lesson 3: Linear differential equations of the first der Solve each of the following differential equations by two methods. Exercise 3.1. Solution. Method 1. It is clear that y + y = 3 e dx = e x is an
More informationSection 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation. is called the reduced equation of (N).
Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1 General Results
More informationPolytechnic Institute of NYU MA 2132 Final Practice Answers Fall 2012
Polytechnic Institute of NYU MA Final Practice Answers Fall Studying from past or sample exams is NOT recommended. If you do, it should be only AFTER you know how to do all of the homework and worksheet
More informationSection 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation
Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1 General Results
More informationChapter 2. First-Order Differential Equations
Chapter 2 First-Order Differential Equations i Let M(x, y) + N(x, y) = 0 Some equations can be written in the form A(x) + B(y) = 0 DEFINITION 2.2. (Separable Equation) A first-order differential equation
More informationHandbook of Ordinary Differential Equations
Handbook of Ordinary Differential Equations Mark Sullivan July, 28 i Contents Preliminaries. Why bother?...............................2 What s so ordinary about ordinary differential equations?......
More informationLecture 19: Solving linear ODEs + separable techniques for nonlinear ODE s
Lecture 19: Solving linear ODEs + separable techniques for nonlinear ODE s Geoffrey Cowles Department of Fisheries Oceanography School for Marine Science and Technology University of Massachusetts-Dartmouth
More informationSection 3.5: Implicit Differentiation
Section 3.5: Implicit Differentiation In the previous sections, we considered the problem of finding the slopes of the tangent line to a given function y = f(x). The idea of a tangent line however is not
More informationImplicit Differentiation and Inverse Trigonometric Functions
Implicit Differentiation an Inverse Trigonometric Functions MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Summer 2018 Explicit vs. Implicit Functions 0.5 1 y 0.0 y 2 0.5 3 4 1.0 0.5
More informationMA22S3 Summary Sheet: Ordinary Differential Equations
MA22S3 Summary Sheet: Ordinary Differential Equations December 14, 2017 Kreyszig s textbook is a suitable guide for this part of the module. Contents 1 Terminology 1 2 First order separable 2 2.1 Separable
More information2.3 Linear Equations 69
2.3 Linear Equations 69 2.3 Linear Equations An equation y = fx,y) is called first-order linear or a linear equation provided it can be rewritten in the special form 1) y + px)y = rx) for some functions
More informationOrdinary Differential Equations (ODEs)
Chapter 13 Ordinary Differential Equations (ODEs) We briefly review how to solve some of the most standard ODEs. 13.1 First Order Equations 13.1.1 Separable Equations A first-order ordinary differential
More informationMath Applied Differential Equations
Math 256 - Applied Differential Equations Notes Basic Definitions and Concepts A differential equation is an equation that involves one or more of the derivatives (first derivative, second derivative,
More informationy0 = F (t0)+c implies C = y0 F (t0) Integral = area between curve and x-axis (where I.e., f(t)dt = F (b) F (a) wheref is any antiderivative 2.
Calulus pre-requisites you must know. Derivative = slope of tangent line = rate. Integral = area between curve and x-axis (where area can be negative). The Fundamental Theorem of Calculus: Suppose f continuous
More informationAPPLIED MATHEMATICS. Part 1: Ordinary Differential Equations. Wu-ting Tsai
APPLIED MATHEMATICS Part 1: Ordinary Differential Equations Contents 1 First Order Differential Equations 3 1.1 Basic Concepts and Ideas................... 4 1.2 Separable Differential Equations................
More informationC.3 First-Order Linear Differential Equations
A34 APPENDIX C Differential Equations C.3 First-Order Linear Differential Equations Solve first-order linear differential equations. Use first-order linear differential equations to model and solve real-life
More informationThe Fundamental Theorem of Calculus: Suppose f continuous on [a, b]. 1.) If G(x) = x. f(t)dt = F (b) F (a) where F is any antiderivative
1 Calulus pre-requisites you must know. Derivative = slope of tangent line = rate. Integral = area between curve and x-axis (where area can be negative). The Fundamental Theorem of Calculus: Suppose f
More informationPower series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) a n z n. n=0
Lecture 22 Power series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) Recall a few facts about power series: a n z n This series in z is centered at z 0. Here z can
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS NON-LINEAR LINEAR (in y) LINEAR W/ CST COEFFs (in y) FIRST- ORDER 4(y ) 2 +x cos y = x 2 4x 2 y + y cos x = x 2 4y + 3y = cos x ORDINARY DIFF EQs SECOND- ORDER
More informationMAP 2302 Midterm 1 Review Solutions 1
MAP 2302 Midterm 1 Review Solutions 1 The exam will cover sections 1.2, 1.3, 2.2, 2.3, 2.4, and 2.6. All topics from this review sheet or from the suggested exercises are fair game. 1 Give explicit solutions
More informationThe Method of Frobenius
The Method of Frobenius Department of Mathematics IIT Guwahati If either p(x) or q(x) in y + p(x)y + q(x)y = 0 is not analytic near x 0, power series solutions valid near x 0 may or may not exist. If either
More informationLecture 7: Differential Equations
Math 94 Professor: Padraic Bartlett Lecture 7: Differential Equations Week 7 UCSB 205 This is the seventh week of the Mathematics Subject Test GRE prep course; here, we review various techniques used to
More informationLecture Notes on Partial Dierential Equations (PDE)/ MaSc 221+MaSc 225
Lecture Notes on Partial Dierential Equations (PDE)/ MaSc 221+MaSc 225 Dr. Asmaa Al Themairi Assistant Professor a a Department of Mathematical sciences, University of Princess Nourah bint Abdulrahman,
More informationMath 4381 / 6378 Symmetry Analysis
Math 438 / 6378 Smmetr Analsis Elementar ODE Review First Order Equations Ordinar differential equations of the form = F(x, ( are called first order ordinar differential equations. There are a variet of
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Basic Terminology A differential equation is an equation that contains an unknown function together with one or more of its derivatives. 1 Examples: 1. y = 2x + cos x 2. dy dt =
More informationChapter 3. Reading assignment: In this chapter we will cover Sections dx 1 + a 0(x)y(x) = g(x). (1)
Chapter 3 3 Introduction Reading assignment: In this chapter we will cover Sections 3.1 3.6. 3.1 Theory of Linear Equations Recall that an nth order Linear ODE is an equation that can be written in the
More informationConsider an ideal pendulum as shown below. l θ is the angular acceleration θ is the angular velocity
1 Second Order Ordinary Differential Equations 1.1 The harmonic oscillator Consider an ideal pendulum as shown below. θ l Fr mg l θ is the angular acceleration θ is the angular velocity A point mass m
More informationMATH 319, WEEK 2: Initial Value Problems, Existence/Uniqueness, First-Order Linear DEs
MATH 319, WEEK 2: Initial Value Problems, Existence/Uniqueness, First-Order Linear DEs 1 Initial-Value Problems We have seen that differential equations can, in general, given rise to multiple solutions.
More information' '-'in.-i 1 'iritt in \ rrivfi pr' 1 p. ru
V X X Y Y 7 VY Y Y F # < F V 6 7»< V q q $ $» q & V 7» Q F Y Q 6 Q Y F & Q &» & V V» Y V Y [ & Y V» & VV & F > V } & F Q \ Q \» Y / 7 F F V 7 7 x» > QX < #» > X >» < F & V F» > > # < q V 6 & Y Y q < &
More informationMATH 124. Midterm 2 Topics
MATH 124 Midterm 2 Topics Anything you ve learned in class (from lecture and homework) so far is fair game, but here s a list of some main topics since the first midterm that you should be familiar with:
More informationMa 530. Special Methods for First Order Equations. Separation of Variables. Consider the equation. M x,y N x,y y 0
Ma 530 Consider the equation Special Methods for First Order Equations Mx, Nx, 0 1 This equation is first order and first degree. The functions Mx, and Nx, are given. Often we write this as Mx, Nx,d 0
More information2 Series Solutions near a Regular Singular Point
McGill University Math 325A: Differential Equations LECTURE 17: SERIES SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS II 1 Introduction Text: Chap. 8 In this lecture we investigate series solutions for the
More informationMath 10C - Fall Final Exam
Math 1C - Fall 217 - Final Exam Problem 1. Consider the function f(x, y) = 1 x 2 (y 1) 2. (i) Draw the level curve through the point P (1, 2). Find the gradient of f at the point P and draw the gradient
More informationMath Reading assignment for Chapter 1: Study Sections 1.1 and 1.2.
Math 3350 1 Chapter 1 Reading assignment for Chapter 1: Study Sections 1.1 and 1.2. 1.1 Material for Section 1.1 An Ordinary Differential Equation (ODE) is a relation between an independent variable x
More information0.1 Problems to solve
0.1 Problems to solve Homework Set No. NEEP 547 Due September 0, 013 DLH Nonlinear Eqs. reducible to first order: 1. 5pts) Find the general solution to the differential equation: y = [ 1 + y ) ] 3/. 5pts)
More informationMA Ordinary Differential Equations
MA 108 - Ordinary Differential Equations Santanu Dey Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 76 dey@math.iitb.ac.in March 21, 2014 Outline of the lecture Second
More informationLECTURE 14: REGULAR SINGULAR POINTS, EULER EQUATIONS
LECTURE 14: REGULAR SINGULAR POINTS, EULER EQUATIONS 1. Regular Singular Points During the past few lectures, we have been focusing on second order linear ODEs of the form y + p(x)y + q(x)y = g(x). Particularly,
More information12d. Regular Singular Points
October 22, 2012 12d-1 12d. Regular Singular Points We have studied solutions to the linear second order differential equations of the form P (x)y + Q(x)y + R(x)y = 0 (1) in the cases with P, Q, R real
More informationNATIONAL ACADEMY DHARMAPURI TRB MATHEMATICS DIFFERENTAL EQUATIONS. Material Available with Question papers CONTACT ,
NATIONAL ACADEMY DHARMAPURI TRB MATHEMATICS DIFFERENTAL EQUATIONS Material Available with Question papers CONTACT 8486 17507, 70108 65319 TEST BACTH SATURDAY & SUNDAY NATIONAL ACADEMY DHARMAPURI http://www.trbtnpsc.com/013/07/trb-questions-and-stu-materials.html
More informationLESSON 25: LAGRANGE MULTIPLIERS OCTOBER 30, 2017
LESSON 5: LAGRANGE MULTIPLIERS OCTOBER 30, 017 Lagrange multipliers is another method of finding minima and maxima of functions of more than one variable. In fact, many of the problems from the last homework
More informationLecture Notes on. Differential Equations. Emre Sermutlu
Lecture Notes on Differential Equations Emre Sermutlu ISBN: Copyright Notice: To my wife Nurten and my daughters İlayda and Alara Contents Preface ix 1 First Order ODE 1 1.1 Definitions.............................
More informationSolving First Order PDEs
Solving Ryan C. Trinity University Partial Differential Equations Lecture 2 Solving the transport equation Goal: Determine every function u(x, t) that solves u t +v u x = 0, where v is a fixed constant.
More informationDefinition of differential equations and their classification. Methods of solution of first-order differential equations
Introduction to differential equations: overview Definition of differential equations and their classification Solutions of differential equations Initial value problems Existence and uniqueness Mathematical
More information7.3 Singular points and the method of Frobenius
284 CHAPTER 7. POWER SERIES METHODS 7.3 Singular points and the method of Frobenius Note: or.5 lectures, 8.4 and 8.5 in [EP], 5.4 5.7 in [BD] While behaviour of ODEs at singular points is more complicated,
More informationOrdinary Differential Equations (ODEs)
c01.tex 8/10/2010 22: 55 Page 1 PART A Ordinary Differential Equations (ODEs) Chap. 1 First-Order ODEs Sec. 1.1 Basic Concepts. Modeling To get a good start into this chapter and this section, quickly
More informationOrdinary differential equations Notes for FYS3140
Ordinary differential equations Notes for FYS3140 Susanne Viefers, Dept of Physics, University of Oslo April 4, 2018 Abstract Ordinary differential equations show up in many places in physics, and these
More informationApplications of First Order Differential Equation
Dr Mansoor Alshehri King Saud University MATH204-Differential Equations Center of Excellence in Learning and Teaching 1 / 39 Orthogonal Trajectories How to Find Orthogonal Trajectories Growth and Decay
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Chapter 1 Introduction and Basic Terminology Most of the phenomena studied in the sciences and engineering involve processes that change with time. For example, it is well known
More informationPDE: The Method of Characteristics Page 1
PDE: The Method of Characteristics Page y u x (x, y) + x u y (x, y) =, () u(, y) = cos y 2. Solution The partial differential equation given can be rewritten as follows: u(x, y) y, x =, (2) where = / x,
More informationGreen Lab. MAXIMA & ODE2. Cheng Ren, Lin. Department of Marine Engineering National Kaohsiung Marine University
Green Lab. 1/20 MAXIMA & ODE2 Cheng Ren, Lin Department of Marine Engineering National Kaohsiung Marine University email: crlin@mail.nkmu.edu.tw Objectives learn MAXIMA learn ODE2 2/20 ODE2 Method First
More informationSeries Solutions of Linear ODEs
Chapter 2 Series Solutions of Linear ODEs This Chapter is concerned with solutions of linear Ordinary Differential Equations (ODE). We will start by reviewing some basic concepts and solution methods for
More informationMa 221 Homework Solutions Due Date: January 24, 2012
Ma Homewk Solutions Due Date: January, 0. pg. 3 #, 3, 6,, 5, 7 9,, 3;.3 p.5-55 #, 3, 5, 7, 0, 7, 9, (Underlined problems are handed in) In problems, and 5, determine whether the given differential equation
More informationMath 4B Notes. Written by Victoria Kala SH 6432u Office Hours: T 12:45 1:45pm Last updated 7/24/2016
Math 4B Notes Written by Victoria Kala vtkala@math.ucsb.edu SH 6432u Office Hours: T 2:45 :45pm Last updated 7/24/206 Classification of Differential Equations The order of a differential equation is the
More informationFirst order Partial Differential equations
First order Partial Differential equations 0.1 Introduction Definition 0.1.1 A Partial Deferential equation is called linear if the dependent variable and all its derivatives have degree one and not multiple
More informationSolutions to Homework # 1 Math 381, Rice University, Fall (x y) y 2 = 0. Part (b). We make a convenient change of variables:
Hildebrand, Ch. 8, # : Part (a). We compute Subtracting, we eliminate f... Solutions to Homework # Math 38, Rice University, Fall 2003 x = f(x + y) + (x y)f (x + y) y = f(x + y) + (x y)f (x + y). x = 2f(x
More informationDIFFERENTIAL EQUATIONS
Mr. Isaac Akpor Adjei (MSc. Mathematics, MSc. Biostats) isaac.adjei@gmail.com April 7, 2017 ORDINARY In many physical situation, equation arise which involve differential coefficients. For example: 1 The
More informationMATH 312 Section 6.2: Series Solutions about Singular Points
MATH 312 Section 6.2: Series Solutions about Singular Points Prof. Jonathan Duncan Walla Walla University Spring Quarter, 2008 Outline 1 Classifying Singular Points 2 The Method of Frobenius 3 Conclusions
More informationIntroduction to Linear Algebra
Introduction to Linear Algebra Linear algebra is the algebra of vectors. In a course on linear algebra you will also learn about the machinery (matrices and reduction of matrices) for solving systems of
More informationThe second-order 1D wave equation
C The second-order D wave equation C. Homogeneous wave equation with constant speed The simplest form of the second-order wave equation is given by: x 2 = Like the first-order wave equation, it responds
More informationSolutions to Section 1.1
Solutions to Section True-False Review: FALSE A derivative must involve some derivative of the function y f(x), not necessarily the first derivative TRUE The initial conditions accompanying a differential
More informationdy dx dx = 7 1 x dx dy = 7 1 x dx e u du = 1 C = 0
1. = 6x = 6x = 6 x = 6 x x 2 y = 6 2 + C = 3x2 + C General solution: y = 3x 2 + C 3. = 7 x = 7 1 x = 7 1 x General solution: y = 7 ln x + C. = e.2x = e.2x = e.2x (u =.2x, du =.2) y = e u 1.2 du = 1 e u
More informationIntroduction of Partial Differential Equations and Boundary Value Problems
Introduction of Partial Differential Equations and Boundary Value Problems 2009 Outline Definition Classification Where PDEs come from? Well-posed problem, solutions Initial Conditions and Boundary Conditions
More informationIntegration by parts Integration by parts is a direct reversal of the product rule. By integrating both sides, we get:
Integration by parts Integration by parts is a direct reversal of the proct rule. By integrating both sides, we get: u dv dx x n sin mx dx (make u = x n ) dx = uv v dx dx When to use integration by parts
More informationMath 2163, Practice Exam II, Solution
Math 63, Practice Exam II, Solution. (a) f =< f s, f t >=< s e t, s e t >, an v v = , so D v f(, ) =< ()e, e > =< 4, 4 > = 4. (b) f =< xy 3, 3x y 4y 3 > an v =< cos π, sin π >=, so
More informationLecture 2: Review of Prerequisites. Table of contents
Math 348 Fall 217 Lecture 2: Review of Prerequisites Disclaimer. As we have a textbook, this lecture note is for guidance and supplement only. It should not be relied on when preparing for exams. In this
More informationMATH 1231 MATHEMATICS 1B Calculus Section 3A: - First order ODEs.
MATH 1231 MATHEMATICS 1B 2010. For use in Dr Chris Tisdell s lectures. Calculus Section 3A: - First order ODEs. Created and compiled by Chris Tisdell S1: What is an ODE? S2: Motivation S3: Types and orders
More informationMath 225 Differential Equations Notes Chapter 1
Math 225 Differential Equations Notes Chapter 1 Michael Muscedere September 9, 2004 1 Introduction 1.1 Background In science and engineering models are used to describe physical phenomena. Often these
More informationCalifornia State University Northridge MATH 280: Applied Differential Equations Midterm Exam 1
California State University Northridge MATH 280: Applied Differential Equations Midterm Exam 1 October 9, 2013. Duration: 75 Minutes. Instructor: Jing Li Student Name: Student number: Take your time to
More information23. Implicit differentiation
23. 23.1. The equation y = x 2 + 3x + 1 expresses a relationship between the quantities x and y. If a value of x is given, then a corresponding value of y is determined. For instance, if x = 1, then y
More informationSeries Solution of Linear Ordinary Differential Equations
Series Solution of Linear Ordinary Differential Equations Department of Mathematics IIT Guwahati Aim: To study methods for determining series expansions for solutions to linear ODE with variable coefficients.
More information1+t 2 (l) y = 2xy 3 (m) x = 2tx + 1 (n) x = 2tx + t (o) y = 1 + y (p) y = ty (q) y =
DIFFERENTIAL EQUATIONS. Solved exercises.. Find the set of all solutions of the following first order differential equations: (a) x = t (b) y = xy (c) x = x (d) x = (e) x = t (f) x = x t (g) x = x log
More information