Welcome to Math 257/316 - Partial Differential Equations

Size: px
Start display at page:

Download "Welcome to Math 257/316 - Partial Differential Equations"

Transcription

1 Welcome to Math 257/316 - Partial Differential Equations Instructor: Mona Rahmani mrahmani@math.ubc.ca Office: Mathematics Building 110 Office hours: Mondays 2-3 pm, Wednesdays and Fridays 1-2 pm. Course webpage: teaching/math / (all assignments and lecture notes will be posted here.) A list of course topics is given in the course outline on the webpage.

2 List of Topics 1. Review of techniques to solve ODEs 2. Series solutions of variable coefficient ODEs 3. Introduction to partial differential equations a. The heat equation b. The wave equation c. Laplace s equation 4. Introduction to numerical methods for PDEs 5. Fourier series and separation of variables. a. The heat equation b. The wave equation c. Laplace s equation 6. Boundary value problems and Sturm-Liouville theory

3 Text book (recommended but not required): Elementary Differential Equations and Boundary Value Problems by Boyce & DiPrima, Etext. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems (4nd Ed), R. Haberman, (Pearson), Partial Differential Equations for Scientists and Engineers (1st Ed), S. Farlow, (Dover), Online resources: Professor Anthony Peirce s course material: ~peirce/ you can find all last terms lecture notes, previous assignments, exams and their solutions on this webpage. Professor Richard Froese s lecture notes: ~rfroese/notes/lecs316.pdf

4 Formula sheet - also posted on the course webpage: You will get this formula sheet on the exams. Please make sure you become familiar with the notation as we progress through the term. Math PDE Formula sheet - final exam Trigonometric and Hyperbolic Function identities sin( ± )=sin cos ± sin cos sin 2 t + cos 2 t =1 cos( ± ) = cos cos sin sin. sin 2 t = 1 2 (1 cos(2t)) sinh( ± ) = sinh cosh ± sinh cosh cosh 2 t sinh 2 t =1 cosh( ± ) = cosh cosh ± sinh sinh. sinh 2 t = 1 2 (cosh(2t) 1) Basic linear ODE s with real coe cients constant coe cients Euler eq ODE ay 00 + by 0 + cy =0 ax 2 y 00 + bxy 0 + cy =0 indicial eq. ar 2 + br + c =0 ar(r 1) + br + c =0 r 1 6= r 2 real y = Ae r1x + Be r 2x y = Ax r 1 + Bx r 2 r 1 = r 2 = r y = Ae rx + Bxe rx y = Ax r + Bx r ln x r = ± iµ e x [A cos(µx)+bsin(µx)] x [A cos(µ ln x )+Bsin(µ ln x )] Series solutions for y 00 + p(x)y 0 + q(x)y =0(?) around x = x 0. Ordinary point x 0 : Two linearly independent solutions of the form: y(x) = P 1 n=0 a n(x x 0 ) n Regular singular point x 0 : Rearrange (?) as: (x x 0 ) 2 y 00 + [(x x 0 )p(x)](x x 0 )y 0 + [(x x 0 ) 2 q(x)]y =0 If r 1 >r 2 are roots of the indicial equation: r(r 1) + br + c = 0 where b = lim x!x 0 (x x 0 )p(x) and c = lim x!x 0 (x x 0 ) 2 q(x) then a solution of (?) is y 1 (x) = P 1 n=0 a n(x x 0 ) n+r 1 where a 0 =1. The second linerly independent solution y 2 is of the form: Case 1: If r 1 r 2 is neither 0 nor a positive integer: 1X y 2 (x) = b n (x x 0 ) n+r 2 where b 0 =1. Case 2: If r 1 r 2 = 0: n=0 y 2 (x) =y 1 (x) ln(x x 0 )+ 1X b n (x x 0 ) n+r 2 for some b 1,b 2... n=1 Case 3: If r 1 r 2 is a positive integer: 1X y 2 (x) =ay 1 (x) ln(x x 0 )+ b n (x x 0 ) n+r 2 where b 0 =1. n=0 Fourier, sine and cosine series Let f(x) be defined in [ L, L]then its Fourier series Ff(x) isa2l-periodic function on R: Ff(x) = a P 1 n=1 a n cos( n x L )+b n sin( n x R L ) where a n = 1 L n x L f(x) cos( L L ) dx and b R n = 1 L n x L f(x) sin( L L ) dx Theorem (Pointwise convergence) If f(x) and f 0 (x) are piecewise continuous, then Ff(x) converges for every x to 1 2 [f(x )+f(x+)]. Parseval s indentity Z 1 L f(x) 2 dx = a 0 2 1X + a n 2 + b n 2. L 2 L n=1 For f(x) defined in [0,L], its cosine and sine series are Cf(x) = a X Sf(x) = 1X n=1 n=1 a n cos( n x L ), b n sin( n x L ), a n = 2 L b n = 2 L Z L D Alembert s solution to the wave equation 0 Z L 0 f(x) cos( n x L ) dx, f(x) sin( n x L ) dx. PDE: u tt = c 2 u xx, 1 <x<1, t>0 IC: u(x, 0) = f(x), u t (x, 0) = g(x). SOLUTION: u(x, t) = 1 2 [f(x + ct)+f(x ct)] + R 1 x+ct 2c x ct g(s)ds Sturm-Liouville Eigenvalue Problems ODE: [p(x)y 0 ] 0 q(x)y + r(x)y =0, a < x < b. BC: 1 y(a)+ 2 y 0 (a) = 0, 1y(b)+ 2 y 0 (b) = 0. Hypothesis: p, p 0, q, r continuous on [a, b]. p(x) > 0 and r(x) > 0 for x 2 [a, b] > > 0. Properties (1) The di erential operator Ly =[p(x)y 0 ] 0 q(x)y is symmetric in the sense that (f, Lg) =(Lf, g) for all f,g satisfying the BC, where (f,g) = R b a f(x)g(x) dx. (2) All eigenvalues are real and can be ordered as 1 < 2 < < n < with n!1as n!1, and each eigenvalue admits a unique (up to a scalar factor) eigenfunction n. (3) Orthogonality: ( m,r n )= R b a m(x) n (x)r(x) dx = 0 if m 6= n. (4) Expansion: If f(x) :[a, b]! R is square integrable, then R 1X b a f(x) n(x)r(x) dx f(x) = n=1 c n n (x), a<x<b,c n = R b a 2 n(x)r(x) dx,n=1, 2,... 1

5 Grades: Final exam: 50%. Students must get at least 35% on the final exam to pass the course. Two in-class midterm exams (each 20%): 40%. There will be no make-up midterms. If you cannot make it to any of the midterms (for a legitimate reason), you must inform me at least two days before the test date. Homework (including Matlab assignments): 10%. Assignment should be submitted at the beginning of the class on the day they are due. No late submission or electronic submission will be accepted. You must submit your assignment at the section you are registered in. The handed in assignment must be your own work.

6 Midterm dates (two in-class midterms): Friday, February 15 Wednesday, March 20

7 Matlab: Some homework include Matlab assignments. The scripts will be provided. However, the assignments may require some modification of the scripts. Matlab is free for all UBC students. See UBC IT services.

8

9

10

11

12

13

14

1 A complete Fourier series solution

1 A complete Fourier series solution Math 128 Notes 13 In this last set of notes I will try to tie up some loose ends. 1 A complete Fourier series solution First here is an example of the full solution of a pde by Fourier series. Consider

More information

Series Solutions Near a Regular Singular Point

Series 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 information

2 Series Solutions near a Regular Singular Point

2 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 information

FINAL EXAM, MATH 353 SUMMER I 2015

FINAL EXAM, MATH 353 SUMMER I 2015 FINAL EXAM, MATH 353 SUMMER I 25 9:am-2:pm, Thursday, June 25 I have neither given nor received any unauthorized help on this exam and I have conducted myself within the guidelines of the Duke Community

More information

MAE/MSE502 Partial Differential Equations in Engineering. Spring 2019 Mon/Wed 6:00-7:15 PM Classroom: CAVC 101

MAE/MSE502 Partial Differential Equations in Engineering. Spring 2019 Mon/Wed 6:00-7:15 PM Classroom: CAVC 101 MAE/MSE502 Partial Differential Equations in Engineering Spring 2019 Mon/Wed 6:00-7:15 PM Classroom: CAVC 101 Instructor: Huei-Ping Huang, hp.huang@asu.edu Office: ERC 359 Office hours: Monday 3-4 PM,

More information

MATH 251 Final Examination August 14, 2015 FORM A. Name: Student Number: Section:

MATH 251 Final Examination August 14, 2015 FORM A. Name: Student Number: Section: MATH 251 Final Examination August 14, 2015 FORM A Name: Student Number: Section: This exam has 11 questions for a total of 150 points. Show all your work! In order to obtain full credit for partial credit

More information

Lecture IX. Definition 1 A non-singular Sturm 1 -Liouville 2 problem consists of a second order linear differential equation of the form.

Lecture IX. Definition 1 A non-singular Sturm 1 -Liouville 2 problem consists of a second order linear differential equation of the form. Lecture IX Abstract When solving PDEs it is often necessary to represent the solution in terms of a series of orthogonal functions. One way to obtain an orthogonal family of functions is by solving a particular

More information

MTH5201 Mathematical Methods in Science and Engineering 1 Fall 2014 Syllabus

MTH5201 Mathematical Methods in Science and Engineering 1 Fall 2014 Syllabus MTH5201 Mathematical Methods in Science and Engineering 1 Fall 2014 Syllabus Instructor: Dr. Aaron Welters; O ce: Crawford Bldg., Room 319; Phone: (321) 674-7202; Email: awelters@fit.edu O ce hours: Mon.

More information

Lecture 4: Frobenius Series about Regular Singular Points

Lecture 4: Frobenius Series about Regular Singular Points Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without explicit written permission from the copyright owner. 1 Lecture 4: Frobenius

More information

كلية العلوم قسم الرياضيات المعادالت التفاضلية العادية

كلية العلوم قسم الرياضيات المعادالت التفاضلية العادية الجامعة اإلسالمية كلية العلوم غزة قسم الرياضيات المعادالت التفاضلية العادية Elementary differential equations and boundary value problems المحاضرون أ.د. رائد صالحة د. فاتن أبو شوقة 1 3 4 5 6 بسم هللا

More information

Bessel s Equation. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Fall Department of Mathematics

Bessel s Equation. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Fall Department of Mathematics Bessel s Equation MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Background Bessel s equation of order ν has the form where ν is a constant. x 2 y + xy

More information

swapneel/207

swapneel/207 Partial differential equations Swapneel Mahajan www.math.iitb.ac.in/ swapneel/207 1 1 Power series For a real number x 0 and a sequence (a n ) of real numbers, consider the expression a n (x x 0 ) n =

More information

Math Partial Differential Equations

Math Partial Differential Equations Math 531 - Partial Differential Equations to Partial Differential Equations Joseph M. Mahaffy, jmahaffy@sdsu.edu Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences

More information

MAE502/MSE502 Partial Differential Equations in Engineering. Spring 2012 Mon/Wed 5:00-6:15 PM

MAE502/MSE502 Partial Differential Equations in Engineering. Spring 2012 Mon/Wed 5:00-6:15 PM MAE502/MSE502 Partial Differential Equations in Engineering Spring 2012 Mon/Wed 5:00-6:15 PM Instructor: Huei-Ping Huang, hp.huang@asu.edu (Huei rhymes with "way") Office: ERC 359 Office hours: Monday

More information

Lecture Notes for MAE 3100: Introduction to Applied Mathematics

Lecture Notes for MAE 3100: Introduction to Applied Mathematics ecture Notes for MAE 31: Introduction to Applied Mathematics Richard H. Rand Cornell University Ithaca NY 14853 rhr2@cornell.edu http://audiophile.tam.cornell.edu/randdocs/ version 17 Copyright 215 by

More information

Series Solutions of Differential Equations

Series 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 information

MATH 412 Fourier Series and PDE- Spring 2010 SOLUTIONS to HOMEWORK 5

MATH 412 Fourier Series and PDE- Spring 2010 SOLUTIONS to HOMEWORK 5 MATH 4 Fourier Series PDE- Spring SOLUTIONS to HOMEWORK 5 Problem (a: Solve the following Sturm-Liouville problem { (xu + λ x u = < x < e u( = u (e = (b: Show directly that the eigenfunctions are orthogonal

More information

MATH-UA 263 Partial Differential Equations Recitation Summary

MATH-UA 263 Partial Differential Equations Recitation Summary MATH-UA 263 Partial Differential Equations Recitation Summary Yuanxun (Bill) Bao Office Hour: Wednesday 2-4pm, WWH 1003 Email: yxb201@nyu.edu 1 February 2, 2018 Topics: verifying solution to a PDE, dispersion

More information

Lecture Notes on. Differential Equations. Emre Sermutlu

Lecture 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 information

MA22S3 Summary Sheet: Ordinary Differential Equations

MA22S3 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 information

Lecture 1: Review of methods to solve Ordinary Differential Equations

Lecture 1: Review of methods to solve Ordinary Differential Equations Introductory lecture notes on Partial Differential Equations - c Anthony Peirce Not to be copied, used, or revised without explicit written permission from the copyright owner 1 Lecture 1: Review of methods

More information

Monday, 6 th October 2008

Monday, 6 th October 2008 MA211 Lecture 9: 2nd order differential eqns Monday, 6 th October 2008 MA211 Lecture 9: 2nd order differential eqns 1/19 Class test next week... MA211 Lecture 9: 2nd order differential eqns 2/19 This morning

More information

MATH 251 Final Examination May 3, 2017 FORM A. Name: Student Number: Section:

MATH 251 Final Examination May 3, 2017 FORM A. Name: Student Number: Section: MATH 5 Final Examination May 3, 07 FORM A Name: Student Number: Section: This exam has 6 questions for a total of 50 points. In order to obtain full credit for partial credit problems, all work must be

More information

MATH 251 Final Examination December 16, 2015 FORM A. Name: Student Number: Section:

MATH 251 Final Examination December 16, 2015 FORM A. Name: Student Number: Section: MATH 5 Final Examination December 6, 5 FORM A Name: Student Number: Section: This exam has 7 questions for a total of 5 points. In order to obtain full credit for partial credit problems, all work must

More information

Fall Math 3410 Name (Print): Solution KEY Practice Exam 2 - November 4 Time Limit: 50 Minutes

Fall Math 3410 Name (Print): Solution KEY Practice Exam 2 - November 4 Time Limit: 50 Minutes Fall 206 - Math 340 Name (Print): Solution KEY Practice Exam 2 - November 4 Time Limit: 50 Minutes This exam contains pages (including this cover page) and 5 problems. Check to see if any pages are missing.

More information

Two special equations: Bessel s and Legendre s equations. p Fourier-Bessel and Fourier-Legendre series. p

Two special equations: Bessel s and Legendre s equations. p Fourier-Bessel and Fourier-Legendre series. p LECTURE 1 Table of Contents Two special equations: Bessel s and Legendre s equations. p. 259-268. Fourier-Bessel and Fourier-Legendre series. p. 453-460. Boundary value problems in other coordinate system.

More information

Ordinary Differential Equations (ODEs)

Ordinary 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 information

MATH 261 MATH 261: Elementary Differential Equations MATH 261 FALL 2005 FINAL EXAM FALL 2005 FINAL EXAM EXAMINATION COVER PAGE Professor Moseley

MATH 261 MATH 261: Elementary Differential Equations MATH 261 FALL 2005 FINAL EXAM FALL 2005 FINAL EXAM EXAMINATION COVER PAGE Professor Moseley MATH 6 MATH 6: Elementary Differential Equations MATH 6 FALL 5 FINAL EXAM FALL 5 FINAL EXAM EXAMINATION COVER PAGE Professor Moseley PRINT NAME ( ) Last Name, First Name MI (What you wish to be called)

More information

Lecture 13: Series Solutions near Singular Points

Lecture 13: Series Solutions near Singular Points Lecture 13: Series Solutions near Singular Points March 28, 2007 Here we consider solutions to second-order ODE s using series when the coefficients are not necessarily analytic. A first-order analogy

More information

Lecture 10. (2) Functions of two variables. Partial derivatives. Dan Nichols February 27, 2018

Lecture 10. (2) Functions of two variables. Partial derivatives. Dan Nichols February 27, 2018 Lecture 10 Partial derivatives Dan Nichols nichols@math.umass.edu MATH 233, Spring 2018 University of Massachusetts February 27, 2018 Last time: functions of two variables f(x, y) x and y are the independent

More information

SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS

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

More information

MATH 251 Final Examination December 16, 2014 FORM A. Name: Student Number: Section:

MATH 251 Final Examination December 16, 2014 FORM A. Name: Student Number: Section: MATH 2 Final Examination December 6, 204 FORM A Name: Student Number: Section: This exam has 7 questions for a total of 0 points. In order to obtain full credit for partial credit problems, all work must

More information

Introduction and preliminaries

Introduction and preliminaries Chapter Introduction and preliminaries Partial differential equations What is a partial differential equation? ODEs Ordinary Differential Equations) have one variable x). PDEs Partial Differential Equations)

More information

Georgia Tech PHYS 6124 Mathematical Methods of Physics I

Georgia Tech PHYS 6124 Mathematical Methods of Physics I Georgia Tech PHYS 612 Mathematical Methods of Physics I Instructor: Predrag Cvitanović Fall semester 2012 Homework Set #5 due October 2, 2012 == show all your work for maximum credit, == put labels, title,

More information

Math Exam 2, October 14, 2008

Math Exam 2, October 14, 2008 Math 96 - Exam 2, October 4, 28 Name: Problem (5 points Find all solutions to the following system of linear equations, check your work: x + x 2 x 3 2x 2 2x 3 2 x x 2 + x 3 2 Solution Let s perform Gaussian

More information

MATH 312 Section 6.2: Series Solutions about Singular Points

MATH 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 information

Math 4263 Homework Set 1

Math 4263 Homework Set 1 Homework Set 1 1. Solve the following PDE/BVP 2. Solve the following PDE/BVP 2u t + 3u x = 0 u (x, 0) = sin (x) u x + e x u y = 0 u (0, y) = y 2 3. (a) Find the curves γ : t (x (t), y (t)) such that that

More information

Physics 250 Green s functions for ordinary differential equations

Physics 250 Green s functions for ordinary differential equations Physics 25 Green s functions for ordinary differential equations Peter Young November 25, 27 Homogeneous Equations We have already discussed second order linear homogeneous differential equations, which

More information

Ma 221 Final Exam Solutions 5/14/13

Ma 221 Final Exam Solutions 5/14/13 Ma 221 Final Exam Solutions 5/14/13 1. Solve (a) (8 pts) Solution: The equation is separable. dy dx exy y 1 y0 0 y 1e y dy e x dx y 1e y dy e x dx ye y e y dy e x dx ye y e y e y e x c The last step comes

More information

The Method of Frobenius

The 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 information

Name: ID.NO: Fall 97. PLEASE, BE NEAT AND SHOW ALL YOUR WORK; CIRCLE YOUR ANSWER. NO NOTES, BOOKS, CALCULATORS, TAPE PLAYERS, or COMPUTERS.

Name: ID.NO: Fall 97. PLEASE, BE NEAT AND SHOW ALL YOUR WORK; CIRCLE YOUR ANSWER. NO NOTES, BOOKS, CALCULATORS, TAPE PLAYERS, or COMPUTERS. MATH 303-2/6/97 FINAL EXAM - Alternate WILKERSON SECTION Fall 97 Name: ID.NO: PLEASE, BE NEAT AND SHOW ALL YOUR WORK; CIRCLE YOUR ANSWER. NO NOTES, BOOKS, CALCULATORS, TAPE PLAYERS, or COMPUTERS. Problem

More information

Mathematical Modeling using Partial Differential Equations (PDE s)

Mathematical Modeling using Partial Differential Equations (PDE s) Mathematical Modeling using Partial Differential Equations (PDE s) 145. Physical Models: heat conduction, vibration. 146. Mathematical Models: why build them. The solution to the mathematical model will

More information

A Brief Review of Elementary Ordinary Differential Equations

A 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 information

Partial Differential Equations for Engineering Math 312, Fall 2012

Partial Differential Equations for Engineering Math 312, Fall 2012 Partial Differential Equations for Engineering Math 312, Fall 2012 Jens Lorenz July 17, 2012 Contents Department of Mathematics and Statistics, UNM, Albuquerque, NM 87131 1 Second Order ODEs with Constant

More information

MATH-3150H-A: Partial Differential Equation 2018WI - Peterborough Campus

MATH-3150H-A: Partial Differential Equation 2018WI - Peterborough Campus MATH-3150H-A: Partial Differential Equation 2018WI - Peterborough Campus Instructor: Instructor: Kenzu Abdella Email Address: kabdella@trentu.ca Phone Number: 705-748-1011 x7327 Office: GSC 339 Office

More information

Sturm-Liouville Theory

Sturm-Liouville Theory More on Ryan C. Trinity University Partial Differential Equations April 19, 2012 Recall: A Sturm-Liouville (S-L) problem consists of A Sturm-Liouville equation on an interval: (p(x)y ) + (q(x) + λr(x))y

More information

Find the Fourier series of the odd-periodic extension of the function f (x) = 1 for x ( 1, 0). Solution: The Fourier series is.

Find the Fourier series of the odd-periodic extension of the function f (x) = 1 for x ( 1, 0). Solution: The Fourier series is. Review for Final Exam. Monday /09, :45-:45pm in CC-403. Exam is cumulative, -4 problems. 5 grading attempts per problem. Problems similar to homeworks. Integration and LT tables provided. No notes, no

More information

MATH 251 Final Examination August 10, 2011 FORM A. Name: Student Number: Section:

MATH 251 Final Examination August 10, 2011 FORM A. Name: Student Number: Section: MATH 251 Final Examination August 10, 2011 FORM A Name: Student Number: Section: This exam has 10 questions for a total of 150 points. In order to obtain full credit for partial credit problems, all work

More information

The Method of Frobenius

The Method of Frobenius The Method of Frobenius R. C. Trinity University Partial Differential Equations April 7, 2015 Motivating example Failure of the power series method Consider the ODE 2xy +y +y = 0. In standard form this

More information

Degree Master of Science in Mathematical Modelling and Scientific Computing Mathematical Methods I Thursday, 12th January 2012, 9:30 a.m.- 11:30 a.m.

Degree Master of Science in Mathematical Modelling and Scientific Computing Mathematical Methods I Thursday, 12th January 2012, 9:30 a.m.- 11:30 a.m. Degree Master of Science in Mathematical Modelling and Scientific Computing Mathematical Methods I Thursday, 12th January 2012, 9:30 a.m.- 11:30 a.m. Candidates should submit answers to a maximum of four

More information

AE 200 Engineering Analysis and Control of Aerospace Systems

AE 200 Engineering Analysis and Control of Aerospace Systems Instructor Info Credit Class Days / Time Office Location: ENG 272C Office Hours: Monday 4:00pm 6:30pm Email: kamran.turkoglu@sjsu.edu 3 units Tuesday, 6:00pm 8:45pm Classroom CL 222 Prerequisites TA: Contact

More information

Ordinary Differential Equations II

Ordinary Differential Equations II Ordinary Differential Equations II February 23 2017 Separation of variables Wave eq. (PDE) 2 u t (t, x) = 2 u 2 c2 (t, x), x2 c > 0 constant. Describes small vibrations in a homogeneous string. u(t, x)

More information

Guide for Ph.D. Area Examination in Applied Mathematics

Guide for Ph.D. Area Examination in Applied Mathematics Guide for Ph.D. Area Examination in Applied Mathematics (for graduate students in Purdue University s School of Mechanical Engineering) (revised Fall 2016) This is a 3 hour, closed book, written examination.

More information

Section 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation

Section 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 information

12d. Regular Singular Points

12d. 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 information

LECTURE 33: NONHOMOGENEOUS HEAT CONDUCTION PROBLEM

LECTURE 33: NONHOMOGENEOUS HEAT CONDUCTION PROBLEM LECTURE 33: NONHOMOGENEOUS HEAT CONDUCTION PROBLEM 1. General Solving Procedure The general nonhomogeneous 1-dimensional heat conduction problem takes the form Eq : [p(x)u x ] x q(x)u + F (x, t) = r(x)u

More information

Solutions to the Sample Problems for the Final Exam UCLA Math 135, Winter 2015, Professor David Levermore

Solutions to the Sample Problems for the Final Exam UCLA Math 135, Winter 2015, Professor David Levermore Solutions to the Sample Problems for the Final Exam UCLA Math 135, Winter 15, Professor David Levermore Every sample problem for the Midterm exam and every problem on the Midterm exam should be considered

More information

Exam Basics. midterm. 1 There will be 9 questions. 2 The first 3 are on pre-midterm material. 3 The next 1 is a mix of old and new material.

Exam Basics. midterm. 1 There will be 9 questions. 2 The first 3 are on pre-midterm material. 3 The next 1 is a mix of old and new material. Exam Basics 1 There will be 9 questions. 2 The first 3 are on pre-midterm material. 3 The next 1 is a mix of old and new material. 4 The last 5 questions will be on new material since the midterm. 5 60

More information

Week 12: Optimisation and Course Review.

Week 12: Optimisation and Course Review. Week 12: Optimisation and Course Review. MA161/MA1161: Semester 1 Calculus. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway November 21-22, 2016 Assignments. Problem

More information

San Jose State University Department of Mechanical and Aerospace Engineering ME 230, Advanced Mechanical Engineering Analysis, Fall 2015

San Jose State University Department of Mechanical and Aerospace Engineering ME 230, Advanced Mechanical Engineering Analysis, Fall 2015 San Jose State University Department of Mechanical and Aerospace Engineering ME 230, Advanced Mechanical Engineering Analysis, Fall 2015 Instructor: Office Location: Younes Shabany TBD Telephone: (408)

More information

Department of Mathematics. MA 108 Ordinary Differential Equations

Department of Mathematics. MA 108 Ordinary Differential Equations Department of Mathematics Indian Institute of Technology, Bombay Powai, Mumbai 476, INDIA. MA 8 Ordinary Differential Equations Autumn 23 Instructor Santanu Dey Name : Roll No : Syllabus and Course Outline

More information

Welcome to Math 102 Section 102

Welcome to Math 102 Section 102 Welcome to Math 102 Section 102 Mingfeng Qiu Sep. 5, 2018 Math 102: Announcements Instructor: Mingfeng Qiu Email: mqiu@math.ubc.ca Course webpage: https://canvas.ubc.ca Check the calendar!!! Sectional

More information

Name: Math Homework Set # 5. March 12, 2010

Name: Math Homework Set # 5. March 12, 2010 Name: Math 4567. Homework Set # 5 March 12, 2010 Chapter 3 (page 79, problems 1,2), (page 82, problems 1,2), (page 86, problems 2,3), Chapter 4 (page 93, problems 2,3), (page 98, problems 1,2), (page 102,

More information

Practice Problems For Test 3

Practice Problems For Test 3 Practice Problems For Test 3 Power Series Preliminary Material. Find the interval of convergence of the following. Be sure to determine the convergence at the endpoints. (a) ( ) k (x ) k (x 3) k= k (b)

More information

7.3 Singular points and the method of Frobenius

7.3 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 information

Physics 6303 Lecture 9 September 17, ct' 2. ct' ct'

Physics 6303 Lecture 9 September 17, ct' 2. ct' ct' Physics 6303 Lecture 9 September 17, 018 LAST TIME: Finished tensors, vectors, 4-vectors, and 4-tensors One last point is worth mentioning although it is not commonly in use. It does, however, build on

More information

Introduction to Sturm-Liouville Theory and the Theory of Generalized Fourier Series

Introduction to Sturm-Liouville Theory and the Theory of Generalized Fourier Series CHAPTER 5 Introduction to Sturm-Liouville Theory and the Theory of Generalized Fourier Series We start with some introductory examples. 5.. Cauchy s equation The homogeneous Euler-Cauchy equation (Leonhard

More information

Practice Problems For Test 3

Practice Problems For Test 3 Practice Problems For Test 3 Power Series Preliminary Material. Find the interval of convergence of the following. Be sure to determine the convergence at the endpoints. (a) ( ) k (x ) k (x 3) k= k (b)

More information

MATH 251 Final Examination December 19, 2012 FORM A. Name: Student Number: Section:

MATH 251 Final Examination December 19, 2012 FORM A. Name: Student Number: Section: MATH 251 Final Examination December 19, 2012 FORM A Name: Student Number: Section: This exam has 17 questions for a total of 150 points. In order to obtain full credit for partial credit problems, all

More information

LECTURE 14: REGULAR SINGULAR POINTS, EULER EQUATIONS

LECTURE 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 information

Math 1071 Final Review Sheet The following are some review questions to help you study. They do not

Math 1071 Final Review Sheet The following are some review questions to help you study. They do not Math 1071 Final Review Sheet The following are some review questions to help you study. They do not They do The exam represent the entirety of what you could be expected to know on the exam; reflect distribution

More information

1 Solutions in cylindrical coordinates: Bessel functions

1 Solutions in cylindrical coordinates: Bessel functions 1 Solutions in cylindrical coordinates: Bessel functions 1.1 Bessel functions Bessel functions arise as solutions of potential problems in cylindrical coordinates. Laplace s equation in cylindrical coordinates

More information

Physics 6303 Lecture 8 September 25, 2017

Physics 6303 Lecture 8 September 25, 2017 Physics 6303 Lecture 8 September 25, 2017 LAST TIME: Finished tensors, vectors, and matrices At the beginning of the course, I wrote several partial differential equations (PDEs) that are used in many

More information

Review for Exam 2. Review for Exam 2.

Review for Exam 2. Review for Exam 2. Review for Exam 2. 5 or 6 problems. No multiple choice questions. No notes, no books, no calculators. Problems similar to homeworks. Exam covers: Regular-singular points (5.5). Euler differential equation

More information

Consider an ideal pendulum as shown below. l θ is the angular acceleration θ is the angular velocity

Consider 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 information

MA22S2 Lecture Notes on Fourier Series and Partial Differential Equations.

MA22S2 Lecture Notes on Fourier Series and Partial Differential Equations. MAS Lecture Notes on Fourier Series and Partial Differential Equations Joe Ó hógáin E-mail: johog@maths.tcd.ie Main Text: Kreyszig; Advanced Engineering Mathematics Other Texts: Nagle and Saff, Zill and

More information

1 Series Solutions Near Regular Singular Points

1 Series Solutions Near Regular Singular Points 1 Series Solutions Near Regular Singular Points All of the work here will be directed toward finding series solutions of a second order linear homogeneous ordinary differential equation: P xy + Qxy + Rxy

More information

Chapter 4. Series Solutions. 4.1 Introduction to Power Series

Chapter 4. Series Solutions. 4.1 Introduction to Power Series Series Solutions Chapter 4 In most sciences one generation tears down what another has built and what one has established another undoes. In mathematics alone each generation adds a new story to the old

More information

CHAPTER 1. Theory of Second Order. Linear ODE s

CHAPTER 1. Theory of Second Order. Linear ODE s A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 2 A COLLECTION OF HANDOUTS ON SCALAR LINEAR ORDINARY

More information

Series Solution of Linear Ordinary Differential Equations

Series 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 information

MA Chapter 10 practice

MA Chapter 10 practice MA 33 Chapter 1 practice NAME INSTRUCTOR 1. Instructor s names: Chen. Course number: MA33. 3. TEST/QUIZ NUMBER is: 1 if this sheet is yellow if this sheet is blue 3 if this sheet is white 4. Sign the scantron

More information

Additional Homework Problems

Additional Homework Problems Additional Homework Problems These problems supplement the ones assigned from the text. Use complete sentences whenever appropriate. Use mathematical terms appropriately. 1. What is the order of a differential

More information

Chapter 3. Reading assignment: In this chapter we will cover Sections dx 1 + a 0(x)y(x) = g(x). (1)

Chapter 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 information

MATH 345 Differential Equations

MATH 345 Differential Equations MATH 345 Differential Equations Spring 2018 Instructor: Time: Dr. Manuela Girotti; office: Weber 223C email: manuela.girotti@colostate.edu Mon-Tue-Wed-Fri 1:00pm-1:50pm Location: Engineering E 206 Office

More information

Nonconstant Coefficients

Nonconstant Coefficients Chapter 7 Nonconstant Coefficients We return to second-order linear ODEs, but with nonconstant coefficients. That is, we consider (7.1) y + p(t)y + q(t)y = 0, with not both p(t) and q(t) constant. The

More information

Equations with regular-singular points (Sect. 5.5).

Equations with regular-singular points (Sect. 5.5). Equations with regular-singular points (Sect. 5.5). Equations with regular-singular points. s: Equations with regular-singular points. Method to find solutions. : Method to find solutions. Recall: The

More information

Math 251 December 14, 2005 Answer Key to Final Exam. 1 18pt 2 16pt 3 12pt 4 14pt 5 12pt 6 14pt 7 14pt 8 16pt 9 20pt 10 14pt Total 150pt

Math 251 December 14, 2005 Answer Key to Final Exam. 1 18pt 2 16pt 3 12pt 4 14pt 5 12pt 6 14pt 7 14pt 8 16pt 9 20pt 10 14pt Total 150pt Name Section Math 51 December 14, 5 Answer Key to Final Exam There are 1 questions on this exam. Many of them have multiple parts. The point value of each question is indicated either at the beginning

More information

ACM 95b/100b Final Exam March 12, 2004 Due March 17, 2004 ACM 95b/100b 3pm in Firestone 307 E. Sterl Phinney (2 pts) Include grading section number

ACM 95b/100b Final Exam March 12, 2004 Due March 17, 2004 ACM 95b/100b 3pm in Firestone 307 E. Sterl Phinney (2 pts) Include grading section number ACM 95b/100b Final Exam March 12, 2004 Due March 17, 2004 ACM 95b/100b 3pm in Firestone 307 E. Sterl Phinney 2 pts) Include grading section number The honor code is in effect. Please follow all of the

More information

MATH 251 Final Examination May 4, 2015 FORM A. Name: Student Number: Section:

MATH 251 Final Examination May 4, 2015 FORM A. Name: Student Number: Section: MATH 251 Final Examination May 4, 2015 FORM A Name: Student Number: Section: This exam has 16 questions for a total of 150 points. In order to obtain full credit for partial credit problems, all work must

More information

Math 353, Fall 2017 Study guide for the final

Math 353, Fall 2017 Study guide for the final Math 353, Fall 2017 Study guide for the final December 12, 2017 Last updated 12/4/17. If something is unclear or incorrect, please let me know so I can update the documents. Updates 12/4: De-emphasized

More information

4r 2 12r + 9 = 0. r = 24 ± y = e 3x. y = xe 3x. r 2 6r + 25 = 0. y(0) = c 1 = 3 y (0) = 3c 1 + 4c 2 = c 2 = 1

4r 2 12r + 9 = 0. r = 24 ± y = e 3x. y = xe 3x. r 2 6r + 25 = 0. y(0) = c 1 = 3 y (0) = 3c 1 + 4c 2 = c 2 = 1 Mathematics MATB44, Assignment 2 Solutions to Selected Problems Question. Solve 4y 2y + 9y = 0 Soln: The characteristic equation is The solutions are (repeated root) So the solutions are and Question 2

More information

Math Assignment 11

Math Assignment 11 Math 2280 - Assignment 11 Dylan Zwick Fall 2013 Section 8.1-2, 8, 13, 21, 25 Section 8.2-1, 7, 14, 17, 32 Section 8.3-1, 8, 15, 18, 24 1 Section 8.1 - Introduction and Review of Power Series 8.1.2 - Find

More information

Math 2930 Worksheet Final Exam Review

Math 2930 Worksheet Final Exam Review Math 293 Worksheet Final Exam Review Week 14 November 3th, 217 Question 1. (* Solve the initial value problem y y = 2xe x, y( = 1 Question 2. (* Consider the differential equation: y = y y 3. (a Find the

More information

Course Syllabus. Math Differential Equations

Course Syllabus. Math Differential Equations Course Syllabus Math 2320- Differential Equations Catalog Description: Ordinary differential equations, including linear equations, systems of equations, equations with variable coefficients, existence

More information

Phys 631 Mathematical Methods of Theoretical Physics Fall 2018

Phys 631 Mathematical Methods of Theoretical Physics Fall 2018 Phys 631 Mathematical Methods of Theoretical Physics Fall 2018 Course information (updated November 10th) Instructor: Joaquín E. Drut. Email: drut at email.unc.edu. Office: Phillips 296 Where and When:

More information

Math 3B: Lecture 1. Noah White. September 23, 2016

Math 3B: Lecture 1. Noah White. September 23, 2016 Math 3B: Lecture 1 Noah White September 23, 2016 Syllabus Take a copy of the syllabus as you walk in or find it online at math.ucla.edu/~noah Class website There are a few places where you will find/receive

More information

Section 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation. is called the reduced equation of (N).

Section 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 information

Math 251 December 14, 2005 Final Exam. 1 18pt 2 16pt 3 12pt 4 14pt 5 12pt 6 14pt 7 14pt 8 16pt 9 20pt 10 14pt Total 150pt

Math 251 December 14, 2005 Final Exam. 1 18pt 2 16pt 3 12pt 4 14pt 5 12pt 6 14pt 7 14pt 8 16pt 9 20pt 10 14pt Total 150pt Math 251 December 14, 2005 Final Exam Name Section There are 10 questions on this exam. Many of them have multiple parts. The point value of each question is indicated either at the beginning of each question

More information

Lecture 19: Heat conduction with distributed sources/sinks

Lecture 19: Heat conduction with distributed sources/sinks Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without explicit written permission from the copyright owner. 1 ecture 19: Heat conduction

More information

Lahore University of Management Sciences. MATH 210 Introduction to Differential Equations

Lahore University of Management Sciences. MATH 210 Introduction to Differential Equations MATH 210 Introduction to Differential Equations Fall 2016-2017 Instructor Room No. Office Hours Email Telephone Secretary/TA TA Office Hours Course URL (if any) Ali Ashher Zaidi ali.zaidi@lums.edu.pk Math.lums.edu.pk/moodle

More information