Some Special Types of First-Order PDEs Solving Cauchy s problem for nonlinear PDEs. MA 201: Partial Differential Equations Lecture - 6
|
|
- Oswin Lawson
- 5 years ago
- Views:
Transcription
1 MA 201: Partial Differential Equations Lecture - 6
2 Example Find a general solution of p 2 x +q 2 y = u. (1) Solution. To find a general solution, we proceed as follows: Step 1: (Computing f x, f y, f u, f p, f q ). Set f = p 2 x +q 2 y u. Then and hence, f x = p 2, f y = q 2, f u = 1, f p = 2px, f q = 2qy, pf p +qf q = 2p 2 x +2q 2 y, (f x +pf u ) = p 2 +p, (f y +qf u ) = q 2 +q.
3 Step 2: (Writing Charpit s equations and finding a solution g(x,y,u,p,q,a)). The Charpit s equations (or auxiliary) equations are = dx f p = dy = f q dx 2px = dy 2qy = du pf p +qf q = dp (f x +pf u ) = du 2(p 2 x +q 2 y) = dp p 2 +p = dq (f y +qf u ) dq q 2 +q From which it follows that = p 2 dx +2pxdp 2p 3 x +2p 2 x 2p 3 x = p 2 dx +2pxdp p 2 x On integrating, we obtain = q2 dy +2qydq q 2 y log(p 2 x) = log(q 2 y)+loga q 2 dy +2qydq 2q 3 y +2q 2 y 2q 3 y = p 2 x = aq 2 y,where a is an arbitrary constant. (2)
4 Step 3: (Solving for p and q). Using (1) and (2), we find that p 2 x +q 2 y = u, p 2 x = aq 2 y = (aq 2 y)+q 2 y = u = q 2 y(1+a) = u [ ] 1/2 = q 2 u = (1+a)y = q = u. (1+a)y and p 2 = aq 2y x = a u (1+a)y [ ] 1/2 au = p =. (1+a)x y x = au (1+a)x
5 Step 4: (Writing du = p(x,y,u,a)dx +q(x,y,u,a)dy and finding its solution). Writing = Integrate to have [ ] 1/2 ] 1/2 au u du = dx +[ dy (1+a)x (1+a)y ( ) 1/2 1+a ( a ) ( ) 1/2dx 1/2 1 du = + dy. u x y [(1+a)u] 1/2 = (ax) 1/2 +(y) 1/2 +b which gives the general solution of equation (1).
6 Equations involving only p and q If the equation is of the form then Charpit s equations take the form f(p,q) = 0, (3) dx = dy du = = dp f p f q pf p +qf q 0 = dq 0. The last two are actually equivalent to dp dt = 0, dq = 0 and hence dt an immediate solution is given by p = a, where a is an arbitrary constant. Substituting p = a in (3), we obtain a relation Thus, we have q = Q(a). u = ax +Q(a)y +b, (4) where b is a constant. Thus, (4) is a general solution of (3).
7 Example Find a general solution of the equation pq = 1. Solution. If p = a, then pq = 1 q = 1/a. In this case, Q(a) = 1/a. From (4), we obtain a general solution as u = ax + y a +b = a 2 x +y au = b, where a and b are arbitrary constants.
8 Equations not involving the independent variables For equation of the type Charpit s equation becomes f(u,p,q) = 0, (5) dx = dy du = = dp = dq. f p f q pf p +qf q pf u qf u From the last two relations, we have dp = dq = dp pf u qf u p = dq q = p = aq, (6) where a is an arbitrary constant.solving (5) and (6) for p and q, we obtain q = Q(a,u) = p = aq(a,u).
9 Now It gives general solution as where b is an arbitrary constant. du = pdx +qdy = du = aq(a,u)dx +Q(a,u)dy = du = Q(a,u)[adx +dy]. du = ax +y +b, (7) Q(a,u) Example Find a general solution of the PDE p 2 u 2 +q 2 = 1. Solution. Putting p = aq in the given PDE, we obtain a 2 q 2 u 2 +q 2 = 1 = q 2 (1+a 2 u 2 ) = 1 = q = (1+a 2 u 2 ) 1/2.
10 Now, p 2 = (1 q 2 )/u 2 = = p 2 a 2 = 1+a 2 u 2 = p = a(1+a 2 u 2 ) 1/2. ( )( ) (1+a 2 u 2 ) u 2 Substituting p and q in du = pdx +qdy, we obtain du = a(1+a 2 u 2 ) 1/2 dx +(1+a 2 u 2 ) 1/2 dy = (1+a 2 u 2 ) 1/2 du = adx +dy 1 { } = au(1+a 2 u 2 ) 1/2 log[au +(1+a 2 u 2 ) 1/2 ] 2a which is the general solution of the given PDE. = ax +y +b,
11 Separable equations A first-order PDE is separable if it can be written in the form h(x,p) = g(y,q). (8) So that f(x,y,u,p,q) = h(x,p) g(y,q). For this type of equation, Charpit s equations become dx = dy du = = dp = dq. h p g q ph p qg q h x g y Consider two relations having only x and p dp h x = dx h p = dp dx + h x h p = 0. (9) Writing (9) in the form h p dp +h x dx = 0, we see that its solution is h(x,p) = a. Similarly, we get g(y,q) = a.determine p and q and solve equation du = pdx +qdy to determine an integral surface.
12 Example Find a general solution of p 2 y(1+x 2 ) = qx 2. Solution. First we write the given PDE in the form It follows that h(x,p) = p2 (1+x 2 ) x 2 = q y p 2 (1+x 2 ) x 2 = a 2 = p = where a is an arbitrary constant. Similarly, q y = a2 = q = a 2 y. Now, the relation du = pdx +qdy yields du = = g(y,q) (separable equation) ax 1+x 2, ax dx 1+x 2 +a2 ydy = u = a 1+x 2 + a2 y 2 +b, 2 where a and b are arbitrary constants, a general solution for the given PDE.
13 Clairaut s equation A first-order PDE is said to be in Clairaut form if it can be written as u = px +qy +g(p,q). (10) Charpit s equations take the form dx = dy = x +g p y +g q du px +qy +pg p +qg q = dp 0 = dq 0. Clearly p = a and q = b. Substituting the values of p and q in (10), we obtain the required general solution u = ax +by +g(a,b).
14 Example Find a general solution of (p +q)(u xp yq) = 1. Solution. The given PDE can be put in the form u = xp +yq + 1 p +q, (11) which is of Clairaut s type. Putting p = a and q = b in (11), a general solution is given by u = ax +by + 1 a+b, where a and b are arbitrary constants.
15 Method of Characteristics Suppose, we need to find an integral surface for equation and equation f(x,y,u,u x,u y ) = 0. (12) g(x,y,u,p,q) = 0 (13) is compatible to the above equation. Then any point (x,y,u,p,q) on the surface g = 0 satisfies the PDE g f p x +f g q y +(pf p +qf q ) g u (f x +pf u ) g p (f y +qf u ) g q = 0.(14) Thus, we have the following system of five ODEs x (t) = f p, y (t) = f q u (t) = pf p +qf q (15) p (t) = {f x +pf u } q (t) = {f y +qf u } These equations are known as the characteristic equations associated with PDE (12).
16 Let us consider the general Cauchy s problem for the first-order PDE F(x,y,u,u x,u y ) = 0 (16) subject to an appropriate initial condition given by an initial curve Γ C 1 : x(0) = x 0 (τ), y(0) = y 0 (τ), u(0) = u 0 (τ). (17) Note: We need initial conditions also for p and q in order to obtain a complete initial value problem for the system (16)-(17). We have p = u(x,y)/ x = p(x,y). For the transformation we have x = x(t), y = y(t), u = u(t), p = p(x(t),y(t)) = p(t), so that p(0) = u 0(τ) x 0 (τ) = p 0(τ) (say). Similarly, we use following notation q(0) = u 0(τ) y 0 (τ) = q 0(τ) (say).
17 Step-by-step method for solving Cauchy problem: Step 1: Find functions p 0 (τ) and q 0 (τ) (if possible) such that F(x 0 (τ),y 0 (τ),u 0 (τ),p 0 (τ),q 0 (τ)) = 0, u 0(τ) = p 0 (τ)x 0(τ)+q 0 (τ)y 0(τ) Here, we have used the fact that u(0) = u(x(0),y(0)) = u 0 (τ) Note that if p 0 (τ) and q 0 (τ) do not exist, then the Cauchy s problem for (16)-(17) has no solution. If there are several choices for (p 0 (τ),q 0 (τ)), then a solution for the Cauchy s problem (16)-(17) exists for each such choice. Step 2: Solve the characteristic system with the given initial conditions x(0) = x 0 (τ), y(0) = y 0 (τ), u(0) = u 0 (τ), p(0) = p 0 (τ), q(0) = q 0 (τ), where p 0 (τ) and q 0 (τ) are the functions given in Step 1.
18 Step 3: Step 2 will give solutions as x = x(t) = x(t,τ), y = y(t) = y(t,τ), u = u(t) = u(t,τ) (18) Here, check whether the transformation (x,y) (t,τ) is invertible or not. Consider the Jacobian J = F p (x 0 (τ),y 0 (τ),u 0 (τ),p 0 (τ),q 0 (τ))y 0(τ) F q (x 0 (τ),y 0 (τ),u 0 (τ),p 0 (τ),q 0 (τ))x 0 (τ) along given curve Γ. If J 0 (generalized transversality condition), the map (x,y) (t,τ) is invertible around Γ. Example Solve the PDE u x u y u = 0 subject to the condition u(x, x) = 1.
19 Solution. Here, we have f(x,y,u,p,q) = pq u. The characteristic system takes the form dx dt = f p = q(t), dp dt = [f x +p(t)f u ] = p(t), dy dt = f q = p(t), du dt = pf p +qf q = 2p(t)q(t), dq dt = [f y +q(t)f u ] = q(t). Note that dp dt = p(t) = p(t) = cet and dq dt = q(t) = q(t) = det, where c and d are arbitrary constants. From the given equation, we have u(t) = p(t)q(t) = cde 2t.
20 The equations for the characteristic curve are x(t) = de t +d 1, y(t) = ce t +c 1, u(t) = cde 2t, p(t) = ce t, q(t) = de t. Writing the initial condition in parametric form, we have x 0 (τ) = τ, y 0 (τ) = τ, u 0 (τ) = 1. Next, we must find p 0 (τ) and q 0 (τ) such that p 0 (τ)q 0 (τ) u 0 (τ) = 0 & 0 = u 0 (τ) = p 0(τ) q 0 (τ), Thus, we have two choices p 0 (τ) = 1 and q 0 (τ) = 1, or p 0 (τ) = 1 and q 0 (τ) = 1. For the choice p 0 (τ) = 1 = p(0) and q 0 (τ) = 1 = q(0), we obtain x = e t 1+τ, y = e t 1 τ, u = e 2t
21 Check generalized transversality condition J = q 0 (τ) p 0 (τ) 0. From the first two equations, we obtain e t = (x +y +2)/2. Thus, u = (x +y +2)2. 4 If we choose p 0 (τ) = 1 and q 0 (τ) = 1, the solution is given by u(x,y) = (x +y 2)2. 4
Compatible Systems and Charpit s Method Charpit s Method Some Special Types of First-Order PDEs. MA 201: Partial Differential Equations Lecture - 5
Compatible Systems and MA 201: Partial Differential Equations Lecture - 5 Compatible Systems and Definition (Compatible systems of first-order PDEs) A system of two first-order PDEs and f(x,y,u,p,q) 0
More informationCompatible Systems and Charpit s Method
MODULE 2: FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 28 Lecture 5 Compatible Systems Charpit s Method In this lecture, we shall study compatible systems of first-order PDEs the Charpit s method for solving
More informationPartial Differential Equations
Partial Differential Equations Lecture Notes Dr. Q. M. Zaigham Zia Assistant Professor Department of Mathematics COMSATS Institute of Information Technology Islamabad, Pakistan ii Contents 1 Lecture 01
More informationTAM3B DIFFERENTIAL EQUATIONS Unit : I to V
TAM3B DIFFERENTIAL EQUATIONS Unit : I to V Unit I -Syllabus Homogeneous Functions and examples Homogeneous Differential Equations Exact Equations First Order Linear Differential Equations Reduction of
More informationMA 201: Partial Differential Equations Lecture - 2
MA 201: Partial Differential Equations Lecture - 2 Linear First-Order PDEs For a PDE f(x,y,z,p,q) = 0, a solution of the type F(x,y,z,a,b) = 0 (1) which contains two arbitrary constants a and b is said
More informationLecture two. January 17, 2019
Lecture two January 17, 2019 We will learn how to solve rst-order linear equations in this lecture. Example 1. 1) Find all solutions satisfy the equation u x (x, y) = 0. 2) Find the solution if we know
More informationChapter 2. First-Order Partial Differential Equations. Prof. D. C. Sanyal
BY Prof. D. C. Sanyal Retired Professor Of Mathematics University Of Kalyani West Bengal, India E-mail : dcs klyuniv@yahoo.com 1 Module-2: Quasi-Linear Equations of First Order 1. Introduction In this
More informationDepartment of mathematics MA201 Mathematics III
Department of mathematics MA201 Mathematics III Academic Year 2015-2016 Model Solutions: Quiz-II (Set - B) 1. Obtain the bilinear transformation which maps the points z 0, 1, onto the points w i, 1, i
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 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 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 informationLecture 5 - Fundamental Theorem for Line Integrals and Green s Theorem
Lecture 5 - Fundamental Theorem for Line Integrals and Green s Theorem Math 392, section C September 14, 2016 392, section C Lect 5 September 14, 2016 1 / 22 Last Time: Fundamental Theorem for Line Integrals:
More informationSolving First Order PDEs
Solving Ryan C. Trinity University Partial Differential Equations January 21, 2014 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 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 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 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 informationdx n a 1(x) dy
HIGHER ORDER DIFFERENTIAL EQUATIONS Theory of linear equations Initial-value and boundary-value problem nth-order initial value problem is Solve: a n (x) dn y dx n + a n 1(x) dn 1 y dx n 1 +... + a 1(x)
More informationPARTIAL DIFFERENTIAL EQUATIONS
PARTIAL DIFFERENTIAL EQUATIONS Lecturer: P.A. Markowich bgmt2008 http://www.damtp.cam.ac.uk/group/apde/teaching/pde_partii.html In addition to the sets of lecture notes written by previous lecturers ([1],[2])
More informationModule 2: First-Order Partial Differential Equations
Module 2: First-Order Partial Differential Equations The mathematical formulations of many problems in science and engineering reduce to study of first-order PDEs. For instance, the study of first-order
More informationFirst Order Partial Differential Equations: a simple approach for beginners
First Order Partial Differential Equations: a simple approach for beginners Phoolan Prasad Department of Mathematics Indian Institute of Science, Bangalore 560 012 E-mail: prasad@math.iisc.ernet.in URL:
More informationThe Theory of Second Order Linear Differential Equations 1 Michael C. Sullivan Math Department Southern Illinois University
The Theory of Second Order Linear Differential Equations 1 Michael C. Sullivan Math Department Southern Illinois University These notes are intended as a supplement to section 3.2 of the textbook Elementary
More informationNATIONAL OPEN UNIVERSITY OF NIGERIA SCHOOL OF SCIENCE AND TECHNOLOGY COURSE CODE: MTH421 COURSE TITLE: ORDINARY DIFFERENTIAL EQUATIONS
MTH 421 NATIONAL OPEN UNIVERSITY OF NIGERIA SCHOOL OF SCIENCE AND TECHNOLOGY COURSE CODE: MTH421 COURSE TITLE: ORDINARY DIFFERENTIAL EQUATIONS MTH 421 ORDINARY DIFFERENTIAL EQUATIONS COURSE WRITER Prof.
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 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 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 informationLecture 13 - Wednesday April 29th
Lecture 13 - Wednesday April 29th jacques@ucsdedu Key words: Systems of equations, Implicit differentiation Know how to do implicit differentiation, how to use implicit and inverse function theorems 131
More informationAn Introduction to Partial Differential Equations
An Introduction to Partial Differential Equations Ryan C. Trinity University Partial Differential Equations Lecture 1 Ordinary differential equations (ODEs) These are equations of the form where: F(x,y,y,y,y,...)
More informationChapter1. Ordinary Differential Equations
Chapter1. Ordinary Differential Equations In the sciences and engineering, mathematical models are developed to aid in the understanding of physical phenomena. These models often yield an equation that
More informationThe first order quasi-linear PDEs
Chapter 2 The first order quasi-linear PDEs The first order quasi-linear PDEs have the following general form: F (x, u, Du) = 0, (2.1) where x = (x 1, x 2,, x 3 ) R n, u = u(x), Du is the gradient of u.
More informationNonconstant 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 information1 First Order Ordinary Differential Equation
1 Ordinary Differential Equation and Partial Differential Equations S. D. MANJAREKAR Department of Mathematics, Loknete Vyankatrao Hiray Mahavidyalaya Panchavati, Nashik (M.S.), India. shrimathematics@gmail.com
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 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 informationPARTIAL DIFFERENTIAL EQUATIONS. Lecturer: D.M.A. Stuart MT 2007
PARTIAL DIFFERENTIAL EQUATIONS Lecturer: D.M.A. Stuart MT 2007 In addition to the sets of lecture notes written by previous lecturers ([1, 2]) the books [4, 7] are very good for the PDE topics in the course.
More informationDifferential Equations Class Notes
Differential Equations Class Notes Dan Wysocki Spring 213 Contents 1 Introduction 2 2 Classification of Differential Equations 6 2.1 Linear vs. Non-Linear.................................. 7 2.2 Seperable
More informationSolutions to Math 53 Math 53 Practice Final
Solutions to Math 5 Math 5 Practice Final 20 points Consider the initial value problem y t 4yt = te t with y 0 = and y0 = 0 a 8 points Find the Laplace transform of the solution of this IVP b 8 points
More informationz x = f x (x, y, a, b), z y = f y (x, y, a, b). F(x, y, z, z x, z y ) = 0. This is a PDE for the unknown function of two independent variables.
Chapter 2 First order PDE 2.1 How and Why First order PDE appear? 2.1.1 Physical origins Conservation laws form one of the two fundamental parts of any mathematical model of Continuum Mechanics. These
More informationMath 2a Prac Lectures on Differential Equations
Math 2a Prac Lectures on Differential Equations Prof. Dinakar Ramakrishnan 272 Sloan, 253-37 Caltech Office Hours: Fridays 4 5 PM Based on notes taken in class by Stephanie Laga, with a few added comments
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 informationChain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics
3.33pt Chain Rule MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Spring 2019 Single Variable Chain Rule Suppose y = g(x) and z = f (y) then dz dx = d (f (g(x))) dx = f (g(x))g (x)
More informationMathematics II. Tutorial 2 First order differential equations. Groups: B03 & B08
Tutorial 2 First order differential equations Groups: B03 & B08 February 1, 2012 Department of Mathematics National University of Singapore 1/15 : First order linear differential equations In this question,
More informationA review of stability and dynamical behaviors of differential equations:
A review of stability and dynamical behaviors of differential equations: scalar ODE: u t = f(u), system of ODEs: u t = f(u, v), v t = g(u, v), reaction-diffusion equation: u t = D u + f(u), x Ω, with boundary
More information2 Linear Differential Equations General Theory Linear Equations with Constant Coefficients Operator Methods...
MA322 Ordinary Differential Equations Wong Yan Loi 2 Contents First Order Differential Equations 5 Introduction 5 2 Exact Equations, Integrating Factors 8 3 First Order Linear Equations 4 First Order Implicit
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 information( ) and then eliminate t to get y(x) or x(y). Not that the
2. First-order linear equations We want the solution (x,y) of () ax, ( y) x + bx, ( y) y = 0 ( ) and b( x,y) are given functions. If a and b are constants, then =F(bx-ay) where ax, y where F is an arbitrary
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 informationUNIVERSITY OF OSLO. Please make sure that your copy of the problem set is complete before you attempt to answer anything.
UNIVERSITY OF OSLO Faculty of mathematics and natural sciences Examination in MAT2440 Differential equations and optimal control theory Day of examination: 11 June 2015 Examination hours: 0900 1300 This
More informationEntrance Exam, Differential Equations April, (Solve exactly 6 out of the 8 problems) y + 2y + y cos(x 2 y) = 0, y(0) = 2, y (0) = 4.
Entrance Exam, Differential Equations April, 7 (Solve exactly 6 out of the 8 problems). Consider the following initial value problem: { y + y + y cos(x y) =, y() = y. Find all the values y such that the
More informationExistence Theory: Green s Functions
Chapter 5 Existence Theory: Green s Functions In this chapter we describe a method for constructing a Green s Function The method outlined is formal (not rigorous) When we find a solution to a PDE by constructing
More informationLecture 9. Systems of Two First Order Linear ODEs
Math 245 - Mathematics of Physics and Engineering I Lecture 9. Systems of Two First Order Linear ODEs January 30, 2012 Konstantin Zuev (USC) Math 245, Lecture 9 January 30, 2012 1 / 15 Agenda General Form
More informationTwo dimensional manifolds
Two dimensional manifolds We are given a real two-dimensional manifold, M. A point of M is denoted X and local coordinates are X (x, y) R 2. If we use different local coordinates, (x, y ), we have x f(x,
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 informationOrdinary Differential Equation Theory
Part I Ordinary Differential Equation Theory 1 Introductory Theory An n th order ODE for y = y(t) has the form Usually it can be written F (t, y, y,.., y (n) ) = y (n) = f(t, y, y,.., y (n 1) ) (Implicit
More informationReview session Midterm 1
AS.110.109: Calculus II (Eng) Review session Midterm 1 Yi Wang, Johns Hopkins University Fall 2018 7.1: Integration by parts Basic integration method: u-sub, integration table Integration By Parts formula
More information(1 + 2y)y = x. ( x. The right-hand side is a standard integral, so in the end we have the implicit solution. y(x) + y 2 (x) = x2 2 +C.
Midterm 1 33B-1 015 October 1 Find the exact solution of the initial value problem. Indicate the interval of existence. y = x, y( 1) = 0. 1 + y Solution. We observe that the equation is separable, and
More informationNov : Lecture 20: Linear Homogeneous and Heterogeneous ODEs
125 Nov. 09 2005: Lecture 20: Linear Homogeneous and Heterogeneous ODEs Reading: Kreyszig Sections: 1.4 (pp:19 22), 1.5 (pp:25 31), 1.6 (pp:33 38) Ordinary Differential Equations from Physical Models In
More informationREFERENCE: CROFT & DAVISON CHAPTER 20 BLOCKS 1-3
IV ORDINARY DIFFERENTIAL EQUATIONS REFERENCE: CROFT & DAVISON CHAPTER 0 BLOCKS 1-3 INTRODUCTION AND TERMINOLOGY INTRODUCTION A differential equation (d.e.) e) is an equation involving an unknown function
More informationLinear Algebra, part 2 Eigenvalues, eigenvectors and least squares solutions
Linear Algebra, part 2 Eigenvalues, eigenvectors and least squares solutions Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2013 Main problem of linear algebra 2: Given
More information2xy 0 y = ln y 0. d(p 2 x p) =0. p = 1 ± p 1+Cx. 1 ± 1+Cx 1+Cx ln
() Consider the di erential equation Find all solutions and the discriminant xy y = ln y Solution: Let F (x, y, p) = xp y ln p, where p = y Let us consider the surface M = {(x, y, p): F (x, y, p) =} in
More informationMathematical Methods - Lecture 9
Mathematical Methods - Lecture 9 Yuliya Tarabalka Inria Sophia-Antipolis Méditerranée, Titane team, http://www-sop.inria.fr/members/yuliya.tarabalka/ Tel.: +33 (0)4 92 38 77 09 email: yuliya.tarabalka@inria.fr
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 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 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 informationSMA 208: Ordinary differential equations I
SMA 208: Ordinary differential equations I First Order differential equations Lecturer: Dr. Philip Ngare (Contacts: pngare@uonbi.ac.ke, Tue 12-2 PM) School of Mathematics, University of Nairobi Feb 26,
More informationLecture Notes in Mathematics. A First Course in Quasi-Linear Partial Differential Equations for Physical Sciences and Engineering Solution Manual
Lecture Notes in Mathematics A First Course in Quasi-Linear Partial Differential Equations for Physical Sciences and Engineering Solution Manual Marcel B. Finan Arkansas Tech University c All Rights Reserved
More informationf dr. (6.1) f(x i, y i, z i ) r i. (6.2) N i=1
hapter 6 Integrals In this chapter we will look at integrals in more detail. We will look at integrals along a curve, and multi-dimensional integrals in 2 or more dimensions. In physics we use these integrals
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 informationBefore you begin read these instructions carefully.
MATHEMATICAL TRIPOS Part IA Friday, 1 June, 2018 1:30 pm to 4:30 pm PAPER 2 Before you begin read these instructions carefully. The examination paper is divided into two sections. Each question in Section
More informationThe integrating factor method (Sect. 1.1)
The integrating factor method (Sect. 1.1) Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Overview
More informationFirst-Order ODE: Separable Equations, Exact Equations and Integrating Factor
First-Order ODE: Separable Equations, Exact Equations and Integrating Factor Department of Mathematics IIT Guwahati REMARK: In the last theorem of the previous lecture, you can change the open interval
More informationOptimal control problems with PDE constraints
Optimal control problems with PDE constraints Maya Neytcheva CIM, October 2017 General framework Unconstrained optimization problems min f (q) q x R n (real vector) and f : R n R is a smooth function.
More informationMath512 PDE Homework 2
Math51 PDE Homework October 11, 009 Exercise 1.3. Solve u = xu x +yu y +(u x+y y/ = 0 with initial conditon u(x, 0 = 1 x. Proof. In this case, we have F = xp + yq + (p + q / z = 0 and Γ parameterized as
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 information= 2e t e 2t + ( e 2t )e 3t = 2e t e t = e t. Math 20D Final Review
Math D Final Review. Solve the differential equation in two ways, first using variation of parameters and then using undetermined coefficients: Corresponding homogenous equation: with characteristic equation
More informationVolumes of Solids of Revolution Lecture #6 a
Volumes of Solids of Revolution Lecture #6 a Sphereoid Parabaloid Hyperboloid Whateveroid Volumes Calculating 3-D Space an Object Occupies Take a cross-sectional slice. Compute the area of the slice. Multiply
More informationSymmetry Reductions of (2+1) dimensional Equal Width. Wave Equation
Authors: Symmetry Reductions of (2+1) dimensional Equal Width 1. Dr. S. Padmasekaran Wave Equation Asst. Professor, Department of Mathematics Periyar University, Salem 2. M.G. RANI Periyar University,
More informationA( x) B( x) C( x) y( x) 0, A( x) 0
3.1 Lexicon Revisited The nonhomogeneous nd Order ODE has the form: d y dy A( x) B( x) C( x) y( x) F( x), A( x) dx dx The homogeneous nd Order ODE has the form: d y dy A( x) B( x) C( x) y( x), A( x) dx
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 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 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 informationMath 5630: Iterative Methods for Systems of Equations Hung Phan, UMass Lowell March 22, 2018
1 Linear Systems Math 5630: Iterative Methods for Systems of Equations Hung Phan, UMass Lowell March, 018 Consider the system 4x y + z = 7 4x 8y + z = 1 x + y + 5z = 15. We then obtain x = 1 4 (7 + y z)
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 informationLecture 31. Basic Theory of First Order Linear Systems
Math 245 - Mathematics of Physics and Engineering I Lecture 31. Basic Theory of First Order Linear Systems April 4, 2012 Konstantin Zuev (USC) Math 245, Lecture 31 April 4, 2012 1 / 10 Agenda Existence
More information4r 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 information9 More on the 1D Heat Equation
9 More on the D Heat Equation 9. Heat equation on the line with sources: Duhamel s principle Theorem: Consider the Cauchy problem = D 2 u + F (x, t), on x t x 2 u(x, ) = f(x) for x < () where f
More informationContinuous Random Variables
1 / 24 Continuous Random Variables Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay February 27, 2013 2 / 24 Continuous Random Variables
More informationEcon 110: Introduction to Economic Theory. 8th Class 2/7/11
Econ 110: Introduction to Economic Theory 8th Class 2/7/11 go over problem answers from last time; no new problems today given you have your problem set to work on; we'll do some problems for these concepts
More informationResearch Article Equivalent Lagrangians: Generalization, Transformation Maps, and Applications
Journal of Applied Mathematics Volume 01, Article ID 86048, 19 pages doi:10.1155/01/86048 Research Article Equivalent Lagrangians: Generalization, Transformation Maps, and Applications N. Wilson and A.
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 informationFourier transforms. c n e inπx. f (x) = Write same thing in an equivalent form, using n = 1, f (x) = l π
Fourier transforms We can imagine our periodic function having periodicity taken to the limits ± In this case, the function f (x) is not necessarily periodic, but we can still use Fourier transforms (related
More informationChapter 3 Second Order Linear Equations
Partial Differential Equations (Math 3303) A Ë@ Õæ Aë áöß @. X. @ 2015-2014 ú GA JË@ É Ë@ Chapter 3 Second Order Linear Equations Second-order partial differential equations for an known function u(x,
More information4 Introduction to First-Order Partial Differential
4 Introduction to First-Order Partial Differential Equations You have already encountered examples of first-order PDEs in Section 3 when we discussed the pure advection mechanism, and mentioning the interesting
More informationMath Lecture 46
Math 2280 - Lecture 46 Dylan Zwick Fall 2013 Today we re going to use the tools we ve developed in the last two lectures to analyze some systems of nonlinear differential equations that arise in simple
More informationPartial Differential Equations
Partial Differential Equations Lecture Notes Erich Miersemann Department of Mathematics Leipzig University Version October, 2015 2 Contents 1 Introduction 9 1.1 Examples............................. 11
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 informationLecture 22: A Population Growth Equation with Diffusion
M3 - ADVANCED ENGINEERING MATHEMATICS Lecture : A Population Gth Equation with Diffusion The logistic population equation studied in ordinary differential equations classes has the form: ( p (t) = rp(t)
More informationMathematics 426 Robert Gross Homework 9 Answers
Mathematics 4 Robert Gross Homework 9 Answers. Suppose that X is a normal random variable with mean µ and standard deviation σ. Suppose that PX > 9 PX
More informationMATH 452. SAMPLE 3 SOLUTIONS May 3, (10 pts) Let f(x + iy) = u(x, y) + iv(x, y) be an analytic function. Show that u(x, y) is harmonic.
MATH 45 SAMPLE 3 SOLUTIONS May 3, 06. (0 pts) Let f(x + iy) = u(x, y) + iv(x, y) be an analytic function. Show that u(x, y) is harmonic. Because f is holomorphic, u and v satisfy the Cauchy-Riemann equations:
More informationSymmetry and Exact Solutions of (2+1)-Dimensional Generalized Sasa Satsuma Equation via a Modified Direct Method
Commun. Theor. Phys. Beijing, China 51 2009 pp. 97 978 c Chinese Physical Society and IOP Publishing Ltd Vol. 51, No., June 15, 2009 Symmetry and Exact Solutions of 2+1-Dimensional Generalized Sasa Satsuma
More informationCHAPTER VIII. TOTAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER AND DEGREE IN THREE VARIABLES, WHICH ARE DERIVABLE FROM A SINGLE PRIMI TIVE.
CHAPTER VIII. TOTAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER AND DEGREE IN THREE VARIABLES, WHICH ARE DERIVABLE FROM A SINGLE PRIMI TIVE. 95. IN this chapter we shall indicate how the integration of an
More information