Series Solutions Near an Ordinary Point
|
|
- Stella Hicks
- 6 years ago
- Views:
Transcription
1 Series Solutions Near an Ordinary Point MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018
2 Ordinary Points (1 of 2) Consider the second order linear homogeneous ODE: P(t)y + Q(t)y + R(t)y = 0 where P, Q, and R are polynomials. Definition A point t 0 such that P(t 0 ) 0 is called an ordinary point. If P(t 0 ) = 0 then t 0 is called a singular point.
3 Ordinary Points (2 of 2) P(t)y + Q(t)y + R(t)y = 0 If t 0 is an ordinary point, then by continuity there exists an interval (a, b) containing t 0 on which P(t) 0 for all t (a, b). Thus the functions p(t) = Q(t) R(t) and q(t) = are defined P(t) P(t) and continuous on (a, b) and the ODE can be written as y + p(t)y + q(t)y = 0. If the initial conditions are y(t 0 ) = y 0 and y (t 0 ) = y 0 then there exists a unique solution to the ODE satisfying the initial conditions.
4 Power Series Solutions (1 of 5) Consider the ODE y 4y = 0 and find a power series solution with positive radius of convergence centered at an ordinary point.
5 Power Series Solutions (1 of 5) Consider the ODE y 4y = 0 and find a power series solution with positive radius of convergence centered at an ordinary point. Let t 0 = 0 be the ordinary point for simplicity and y(t) = a n t n
6 Power Series Solutions (1 of 5) Consider the ODE y 4y = 0 and find a power series solution with positive radius of convergence centered at an ordinary point. Let t 0 = 0 be the ordinary point for simplicity and y(t) = y (t) = y (t) = a n t n na n t n 1 n=1 n(n 1)a n t n 2. n=2 Substitute into the ODE.
7 Power Series Solutions (2 of 5) 0 = y 4y
8 Power Series Solutions (2 of 5) 0 = y 4y = n(n 1)a n t n 2 4 a n t n n=2
9 Power Series Solutions (2 of 5) 0 = y 4y = n(n 1)a n t n 2 4 a n t n = = n=2 (n + 2)(n + 1)a n+2 t n 4 a n t n [a n+2 (n + 2)(n + 1) 4a n ] t n
10 Power Series Solutions (2 of 5) 0 = y 4y = n(n 1)a n t n 2 4 a n t n = = n=2 (n + 2)(n + 1)a n+2 t n 4 a n t n [a n+2 (n + 2)(n + 1) 4a n ] t n 0 = a n+2 (n + 2)(n + 1) 4a n (for n = 0, 1, 2,...) a n+2 = 2 2 a n (n + 2)(n + 1) This last equation is called a recurrence relation.
11 Power Series Solutions (3 of 5) Let a 0 be arbitrary, then a 2 = a 4 = a 6 = 2 2 (2)(1) a 0 = 22 2! a (4)(3) a 2 = 24 4! a (6)(5) a 4 = 26 6! a 0. a 2n = 22n (2n)! a 0.
12 Power Series Solutions (4 of 5) Let a 1 be arbitrary, then a 3 = a 5 = a 7 =. a 2n+1 = 2 2 (3)(2) a 1 = 22 3! a (5)(4) a 3 = 24 5! a (7)(6) a 5 = 26 7! a 1 2 2n (2n + 1)! a 1.
13 Power Series Solutions (5 of 5) Thus the general solution to y 4y = 0 can be written as y(t) = a 2n t 2n + a 2n+1 t 2n+1 = a 0 = a 0 2 2n (2n)! t2n + a 1 (2t) 2n (2n)! + a 1 2 = a 0 cosh(2t) + a 1 2 sinh(2t) 2 2n (2n + 1)! t2n+1 (2t) 2n+1 (2n + 1)! We can confirm this series converges for all t R.
14 Airy s Equation (1 of 6) Find a power series solution about the ordinary point t 0 = 0 to Airy s equation: y t y = 0.
15 Airy s Equation (1 of 6) Find a power series solution about the ordinary point t 0 = 0 to Airy s equation: y t y = 0. y(t) = y (t) = y (t) = a n t n na n t n 1 n=1 n(n 1)a n t n 2. n=2 Substitute into the ODE.
16 Airy s Equation (2 of 6) 0 = y ty = n(n 1)a n t n 2 t a n t n = n=2 (n + 2)(n + 1)a n+2 t n a n t n+1 = 2a 2 + (n + 2)(n + 1)a n+2 t n a n 1 t n n=1 n=1 = 2a 2 + [(n + 2)(n + 1)a n+2 a n 1 ] t n n=1
17 Airy s Equation (3 of 6) Thus 0 = 2a 2 0 = (n + 2)(n + 1)a n+2 a n 1 (for n = 1, 2,....) From the last equation we obtain the recurrence relation: a n+2 = a n 1 (n + 2)(n + 1).
18 Airy s Equation (3 of 6) Thus 0 = 2a 2 0 = (n + 2)(n + 1)a n+2 a n 1 (for n = 1, 2,....) From the last equation we obtain the recurrence relation: a n+2 = a n 1 (n + 2)(n + 1). Since a 2 = 0 then a 5 = a 8 = a 11 = = 0.
19 Airy s Equation (4 of 6) Let a 0 be arbitrary then a 3 = a a 6 = a = a a 9 = a = a a 3n = a 0 (2)(3)(5)(6) (3n 4)(3n 3)(3n 1)(3n).
20 Airy s Equation (5 of 6) Let a 1 be arbitrary then a 4 = a a 7 = a = a a 7 a 10 = 9 10 = a a 3n+1 = a 1 (3)(4)(6)(7) (3n 3)(3n 2)(3n)(3n + 1).
21 Airy s Equation (6 of 6) Thus the solution to Airy s equation is y(t) = a 0 [1 + n=1 + a 1 [t + y ty = 0 ] t 3n (2)(3)(5)(6) (3n 4)(3n 3)(3n 1)(3n) n=1 = a 0 y 1 (t) + a 1 y 2 (t). t 3n+1 (3)(4)(6)(7) (3n 3)(3n 2)(3n)(3n + 1) ]
22 Illustration y x -5 y 1 (t) y 2 (t) -10
23 Homework Read Section 5.2 Exercises: 1, 2, 3, 5, 6, 21
24 Review We have determined a method for solving an ODE of the form P(t)y + Q(t)y + R(t)y = 0 where P, Q, and R are polynomials. The solution is a power series of the form y(t) = a n (t t 0 ) n where t 0 is an ordinary point.
25 Review We have determined a method for solving an ODE of the form P(t)y + Q(t)y + R(t)y = 0 where P, Q, and R are polynomials. The solution is a power series of the form y(t) = where t 0 is an ordinary point. a n (t t 0 ) n Today we extend this work to a broader range of functions than polynomial P, Q, and R.
26 Differentiation of Power Series Suppose φ(t) = a n (t t 0 ) n has a positive radius of convergence, then we may differentiate the series term-by-term. n φ (n) (t) φ (n) (t 0 ) 0 a n (t t 0 ) n a m na n (t t 0 ) n 1 a 1 n=1 n(n 1)a n (t t 0 ) n 2 2a 2 n=2. n(n 1) (n m + 1)a n (t t 0 ) n m n=m. m!a m
27 Solution to an ODE Suppose φ(t) solves the ODE y + p(t)y + q(t)y = 0 then 0 = φ (t) + p(t)φ (t) + q(t)φ(t) φ (t) = p(t)φ (t) q(t)φ(t)
28 Solution to an ODE Suppose φ(t) solves the ODE y + p(t)y + q(t)y = 0 then 0 = φ (t) + p(t)φ (t) + q(t)φ(t) φ (t) = p(t)φ (t) q(t)φ(t) φ (t 0 ) = p(t 0 )φ (t 0 ) q(t 0 )φ(t 0 )
29 Solution to an ODE Suppose φ(t) solves the ODE y + p(t)y + q(t)y = 0 then 0 = φ (t) + p(t)φ (t) + q(t)φ(t) φ (t) = p(t)φ (t) q(t)φ(t) φ (t 0 ) = p(t 0 )φ (t 0 ) q(t 0 )φ(t 0 ) 2!a 2 = p(t 0 )a 1 q(t 0 )a 0.
30 Solution to an ODE Suppose φ(t) solves the ODE y + p(t)y + q(t)y = 0 then 0 = φ (t) + p(t)φ (t) + q(t)φ(t) φ (t) = p(t)φ (t) q(t)φ(t) φ (t 0 ) = p(t 0 )φ (t 0 ) q(t 0 )φ(t 0 ) 2!a 2 = p(t 0 )a 1 q(t 0 )a 0. Thus we may find a 2 in terms of a 0 and a 1.
31 Continued Differentiation (1 of 2) With patience we can find higher order terms in the series solution through continued differentiation. φ (t) = p(t)φ (t) q(t)φ(t) φ (t) = p (t)φ (t) p(t)φ (t) q (t)φ(t) q(t)φ (t)
32 Continued Differentiation (1 of 2) With patience we can find higher order terms in the series solution through continued differentiation. φ (t) = p(t)φ (t) q(t)φ(t) φ (t) = p (t)φ (t) p(t)φ (t) q (t)φ(t) q(t)φ (t) φ (t 0 ) = p (t 0 )φ (t 0 ) p(t 0 )φ (t 0 ) q (t 0 )φ(t 0 ) q(t 0 )φ (t 0 ) 3!a 3 = p (t 0 )a 1 2p(t 0 )a 2 q (t 0 )a 0 q(t 0 )a 1
33 Continued Differentiation (1 of 2) With patience we can find higher order terms in the series solution through continued differentiation. φ (t) = p(t)φ (t) q(t)φ(t) φ (t) = p (t)φ (t) p(t)φ (t) q (t)φ(t) q(t)φ (t) φ (t 0 ) = p (t 0 )φ (t 0 ) p(t 0 )φ (t 0 ) q (t 0 )φ(t 0 ) q(t 0 )φ (t 0 ) 3!a 3 = p (t 0 )a 1 2p(t 0 )a 2 q (t 0 )a 0 q(t 0 )a 1 Substituting the previously determined value of a 2 we may find a 3 in terms of a 0 and a 1.
34 Continued Differentiation (2 of 2) We can proceed by repeated differentiation to find a 4, a 5,... provided:
35 Continued Differentiation (2 of 2) We can proceed by repeated differentiation to find a 4, a 5,... provided: p(t) and q(t) have derivatives of all orders, and
36 Continued Differentiation (2 of 2) We can proceed by repeated differentiation to find a 4, a 5,... provided: p(t) and q(t) have derivatives of all orders, and we can show the resulting power series converges.
37 Example Assuming that y = φ(t) is a solution to the IVP: y + t 2 y + (sin t)y = 0 y(0) = 1 y (0) = 1 find the first four nonzero terms in the power series representation of φ(t).
38 Example Assuming that y = φ(t) is a solution to the IVP: y + t 2 y + (sin t)y = 0 y(0) = 1 y (0) = 1 find the first four nonzero terms in the power series representation of φ(t). a 0 = 1 a 1 = 1 a 2 = 0 a 3 = 1 3! a 4 = 1 3!
39 Analytic Functions If p(t) and q(t) are analytic functions at t 0, in other words have Taylor series expansions about t 0 which converge to p(t) and q(t) respectively then p and q will have derivatives of all orders at t 0. p(t) = q(t) = p n (t t 0 ) n q n (t t 0 ) n
40 Ordinary and Singular Points Revisited Suppose P(t)y + Q(t)y + R(t)y = 0 y + Q(t) P(t) y + R(t) P(t) y = 0 y + p(t)y + q(t)y = 0 where p(t) = Q(t) P(t) and q(t) = R(t) P(t)
41 Ordinary and Singular Points Revisited Suppose where P(t)y + Q(t)y + R(t)y = 0 y + Q(t) P(t) y + R(t) P(t) y = 0 y + p(t)y + q(t)y = 0 p(t) = Q(t) P(t) and q(t) = R(t) P(t) If p(t) and q(t) are analytic at t 0, then we say that t 0 is an ordinary point of the ODE. Otherwise t 0 is a singular point.
42 Main Result Theorem If t 0 is an ordinary point of the ODE P(t)y + Q(t)y + R(t)y = 0 then the general solution of the ODE is y(t) = a n (t t 0 ) n = a 0 y 1 (t) + a 1 y 2 (t) where a 0 and a 1 are arbitrary and y 1 and y 2 are linearly independent series solutions that are analytic at t 0. Further the radius of convergence for each of y 1 and y 2 is at least as large as the minimum of the radii of convergence for the series p(t) = Q(t)/P(t) and q(t) = R(t)/P(t).
43 Radius of Convergence If p(t) = Q(t)/P(t) and q(t) = R(t)/P(t) and p(t) and q(t) are analytic at t 0 then from the theory of complex variables we have the result that the radius of convergence of p(t) (and similarly q(t)) is at least as large as the minimum distance from t 0 to any root of P(t) in the complex plane.
44 Example The value t 0 = 1 is an ordinary point of the ODE t 2 y + (1 + t)y + 3(ln t)y = 0. Find the radius of convergence of p(t) = 1 + t t 2 q(t) = 3 ln t t 2. and
45 Example The value t 0 = 0 is an ordinary point of the ODE (1 + t 4 )y + 4ty + y = 0. Find the radius of convergence of p(t) = q(t) = t 4. 4t 1 + t 4 and
46 Example The value t 0 = 0 is an ordinary point of the ODE (1 + t 4 )y + 4ty + y = 0. Find the radius of convergence of p(t) = q(t) = t 4. Using Euler s Identity: t = 0 t 4 = 1 = e i(2n+1)π t = e i(2n+1)π/4 2 2 t = ± 2 ± i 2. 4t 1 + t 4 and
47 Illustration 1.0 i y x -0.5
48 Example The value t 0 = 1 is an ordinary point of the ODE (1 + t 4 )y + 4ty + y = 0. Find the radius of convergence of p(t) = q(t) = t 4. 4t 1 + t 4 and
49 Illustration 1.0 i y x -0.5
50 Homework Read Section 5.3 Exercises: 1 7 odd, 10, 11, 22 29
Linear Independence and the Wronskian
Linear Independence and the Wronskian MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Operator Notation Let functions p(t) and q(t) be continuous functions
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 informationHomogeneous Equations with Constant Coefficients
Homogeneous Equations with Constant Coefficients MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 General Second Order ODE Second order ODEs have the form
More informationReview of Power Series
Review of Power Series MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Introduction In addition to the techniques we have studied so far, we may use power
More informationMATH 308 Differential Equations
MATH 308 Differential Equations Summer, 2014, SET 5 JoungDong Kim Set 5: Section 3.1, 3.2 Chapter 3. Second Order Linear Equations. Section 3.1 Homogeneous Equations with Constant Coefficients. In this
More informationCalculus C (ordinary differential equations)
Calculus C (ordinary differential equations) Lesson 9: Matrix exponential of a symmetric matrix Coefficient matrices with a full set of eigenvectors Solving linear ODE s by power series Solutions to linear
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 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 informationLinear Independence. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics
Linear Independence MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Given a set of vectors {v 1, v 2,..., v r } and another vector v span{v 1, v 2,...,
More informationChapter 4: Higher Order Linear Equations
Chapter 4: Higher Order Linear Equations MATH 351 California State University, Northridge April 7, 2014 MATH 351 (Differential Equations) Ch 4 April 7, 2014 1 / 11 Sec. 4.1: General Theory of nth Order
More informationBessel 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 informationTaylor and Maclaurin Series
Taylor and Maclaurin Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Background We have seen that some power series converge. When they do, we can think of them as
More informationSecond Order Linear Equations
October 13, 2016 1 Second And Higher Order Linear Equations In first part of this chapter, we consider second order linear ordinary linear equations, i.e., a differential equation of the form L[y] = d
More informationMathematical Models. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Spring Department of Mathematics
Mathematical Models MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Ordinary Differential Equations The topic of ordinary differential equations (ODEs)
More informationMathematical Models. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Fall Department of Mathematics
Mathematical Models MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Ordinary Differential Equations The topic of ordinary differential equations (ODEs) is
More informationReview 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 informationDifferential Equations Practice: Euler Equations & Regular Singular Points Page 1
Differential Equations Practice: Euler Equations & Regular Singular Points Page 1 Questions Eample (5.4.1) Determine the solution to the differential equation y + 4y + y = 0 that is valid in any interval
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 informationLecture 16. Theory of Second Order Linear Homogeneous ODEs
Math 245 - Mathematics of Physics and Engineering I Lecture 16. Theory of Second Order Linear Homogeneous ODEs February 17, 2012 Konstantin Zuev (USC) Math 245, Lecture 16 February 17, 2012 1 / 12 Agenda
More informationSection 5.2 Series Solution Near Ordinary Point
DE Section 5.2 Series Solution Near Ordinary Point Page 1 of 5 Section 5.2 Series Solution Near Ordinary Point We are interested in second order homogeneous linear differential equations with variable
More informationSystems of Ordinary Differential Equations
Systems of Ordinary Differential Equations MATH 365 Ordinary Differential Equations J Robert Buchanan Department of Mathematics Fall 2018 Objectives Many physical problems involve a number of separate
More informationMath 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 informationHomework 9 - Solutions. Math 2177, Lecturer: Alena Erchenko
Homework 9 - Solutions Math 2177, Lecturer: Alena Erchenko 1. Classify the following differential equations (order, determine if it is linear or nonlinear, if it is linear, then determine if it is homogeneous
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 informationA: Brief Review of Ordinary Differential Equations
A: Brief Review of Ordinary Differential Equations Because of Principle # 1 mentioned in the Opening Remarks section, you should review your notes from your ordinary differential equations (odes) course
More information144 Chapter 3. Second Order Linear Equations
144 Chapter 3. Second Order Linear Equations PROBLEMS In each of Problems 1 through 8 find the general solution of the given differential equation. 1. y + 2y 3y = 0 2. y + 3y + 2y = 0 3. 6y y y = 0 4.
More informationExam 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 informationSecond-Order Linear ODEs
Second-Order Linear ODEs A second order ODE is called linear if it can be written as y + p(t)y + q(t)y = r(t). (0.1) It is called homogeneous if r(t) = 0, and nonhomogeneous otherwise. We shall assume
More informationFirst and Second Order Differential Equations Lecture 4
First and Second Order Differential Equations Lecture 4 Dibyajyoti Deb 4.1. Outline of Lecture The Existence and the Uniqueness Theorem Homogeneous Equations with Constant Coefficients 4.2. The Existence
More informationChapter 2: First Order DE 2.4 Linear vs. Nonlinear DEs
Chapter 2: First Order DE 2.4 Linear vs. Nonlinear DEs First Order DE 2.4 Linear vs. Nonlinear DE We recall the general form of the First Oreder DEs (FODE): dy = f(t, y) (1) dt where f(t, y) is a function
More informationExam II Review: Selected Solutions and Answers
November 9, 2011 Exam II Review: Selected Solutions and Answers NOTE: For additional worked problems see last year s review sheet and answers, the notes from class, and your text. Answers to problems from
More informationSeparable Differential Equations
Separable Differential Equations MATH 6 Calculus I J. Robert Buchanan Department of Mathematics Fall 207 Background We have previously solved differential equations of the forms: y (t) = k y(t) (exponential
More informationSOLUTIONS ABOUT ORDINARY POINTS
238 CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS In Problems 23 and 24 use a substitution to shift the summation index so that the general term of given power series involves x k. 23. nc n x n2 n 24.
More informationElementary Matrices. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics
Elementary Matrices MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Outline Today s discussion will focus on: elementary matrices and their properties, using elementary
More informationFirst Order Systems of Linear Equations. or ODEs of Arbitrary Order
First Order Systems of Linear Equations or ODEs of Arbitrary Order Systems of Equations Relate Quantities Examples Predator-Prey Relationships r 0 = r (100 f) f 0 = f (r 50) (Lokta-Volterra Model) Systems
More informationLecture 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 informationODE Homework Series Solutions Near an Ordinary Point, Part I 1. Seek power series solution of the equation. n(n 1)a n x n 2 = n=0
ODE Homework 6 5.2. Series Solutions Near an Ordinary Point, Part I 1. Seek power series solution of the equation y + k 2 x 2 y = 0, k a constant about the the point x 0 = 0. Find the recurrence relation;
More informationAbsolute Convergence and the Ratio Test
Absolute Convergence and the Ratio Test MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Bacground Remar: All previously covered tests for convergence/divergence apply only
More informationLecture Notes for Math 251: ODE and PDE. Lecture 12: 3.3 Complex Roots of the Characteristic Equation
Lecture Notes for Math 21: ODE and PDE. Lecture 12: 3.3 Complex Roots of the Characteristic Equation Shawn D. Ryan Spring 2012 1 Complex Roots of the Characteristic Equation Last Time: We considered the
More information6. Linear Differential Equations of the Second Order
September 26, 2012 6-1 6. Linear Differential Equations of the Second Order A differential equation of the form L(y) = g is called linear if L is a linear operator and g = g(t) is continuous. The most
More informationWork sheet / Things to know. Chapter 3
MATH 251 Work sheet / Things to know 1. Second order linear differential equation Standard form: Chapter 3 What makes it homogeneous? We will, for the most part, work with equations with constant coefficients
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 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 informationMethod of Frobenius. General Considerations. L. Nielsen, Ph.D. Dierential Equations, Fall Department of Mathematics, Creighton University
Method of Frobenius General Considerations L. Nielsen, Ph.D. Department of Mathematics, Creighton University Dierential Equations, Fall 2008 Outline 1 The Dierential Equation and Assumptions 2 3 Main Theorem
More informationMath 334 A1 Homework 3 (Due Nov. 5 5pm)
Math 334 A1 Homework 3 Due Nov. 5 5pm No Advanced or Challenge problems will appear in homeworks. Basic Problems Problem 1. 4.1 11 Verify that the given functions are solutions of the differential equation,
More informationSign the pledge. On my honor, I have neither given nor received unauthorized aid on this Exam : 11. a b c d e. 1. a b c d e. 2.
Math 258 Name: Final Exam Instructor: May 7, 2 Section: Calculators are NOT allowed. Do not remove this answer page you will return the whole exam. You will be allowed 2 hours to do the test. You may leave
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 informationSolutions for homework 1. 1 Introduction to Differential Equations
Solutions for homework 1 1 Introduction to Differential Equations 1.1 Differential Equation Models The phrase y is proportional to x implies that y is related to x via the equation y = kx, where k is a
More informationOn linear and non-linear equations.(sect. 2.4).
On linear and non-linear equations.sect. 2.4). Review: Linear differential equations. Non-linear differential equations. Properties of solutions to non-linear ODE. The Bernoulli equation. Review: Linear
More informationElementary Differential Equations, Section 2 Prof. Loftin: Practice Test Problems for Test Find the radius of convergence of the power series
Elementary Differential Equations, Section 2 Prof. Loftin: Practice Test Problems for Test 2 SOLUTIONS 1. Find the radius of convergence of the power series Show your work. x + x2 2 + x3 3 + x4 4 + + xn
More information2.1 Differential Equations and Solutions. Blerina Xhabli
2.1 Math 3331 Differential Equations 2.1 Differential Equations and Solutions Blerina Xhabli Department of Mathematics, University of Houston blerina@math.uh.edu math.uh.edu/ blerina/teaching.html Blerina
More informationSeries Solutions. 8.1 Taylor Polynomials
8 Series Solutions 8.1 Taylor Polynomials Polynomial functions, as we have seen, are well behaved. They are continuous everywhere, and have continuous derivatives of all orders everywhere. It also turns
More informationJuly 21 Math 2254 sec 001 Summer 2015
July 21 Math 2254 sec 001 Summer 2015 Section 8.8: Power Series Theorem: Let a n (x c) n have positive radius of convergence R, and let the function f be defined by this power series f (x) = a n (x c)
More informationMath53: Ordinary Differential Equations Autumn 2004
Math53: Ordinary Differential Equations Autumn 2004 Unit 2 Summary Second- and Higher-Order Ordinary Differential Equations Extremely Important: Euler s formula Very Important: finding solutions to linear
More informationMath Ordinary Differential Equations
Math 411 - Ordinary Differential Equations Review Notes - 1 1 - Basic Theory A first order ordinary differential equation has the form x = f(t, x) (11) Here x = dx/dt Given an initial data x(t 0 ) = x
More informationLecture Notes for Math 251: ODE and PDE. Lecture 7: 2.4 Differences Between Linear and Nonlinear Equations
Lecture Notes for Math 51: ODE and PDE. Lecture 7:.4 Differences Between Linear and Nonlinear Equations Shawn D. Ryan Spring 01 1 Existence and Uniqueness Last Time: We developed 1st Order ODE models for
More informationRow Space, Column Space, and Nullspace
Row Space, Column Space, and Nullspace MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Every matrix has associated with it three vector spaces: row space
More informationSection 4.7: Variable-Coefficient Equations
Cauchy-Euler Equations Section 4.7: Variable-Coefficient Equations Before concluding our study of second-order linear DE s, let us summarize what we ve done. In Sections 4.2 and 4.3 we showed how to find
More informationMATH 308 Differential Equations
MATH 308 Differential Equations Summer, 2014, SET 6 JoungDong Kim Set 6: Section 3.3, 3.4, 3.5, 3.6 Section 3.3 Complex Roots of the Characteristic Equation Recall that a second order ODE with constant
More informationLecture Notes for Math 251: ODE and PDE. Lecture 13: 3.4 Repeated Roots and Reduction Of Order
Lecture Notes for Math 251: ODE and PDE. Lecture 13: 3.4 Repeated Roots and Reduction Of Order Shawn D. Ryan Spring 2012 1 Repeated Roots of the Characteristic Equation and Reduction of Order Last Time:
More informationFirst Order ODEs (cont). Modeling with First Order ODEs
First Order ODEs (cont). Modeling with First Order ODEs September 11 15, 2017 Bernoulli s ODEs Yuliya Gorb Definition A first order ODE is called a Bernoulli s equation iff it is written in the form y
More informationNumerical Methods - Boundary Value Problems for ODEs
Numerical Methods - Boundary Value Problems for ODEs Y. K. Goh Universiti Tunku Abdul Rahman 2013 Y. K. Goh (UTAR) Numerical Methods - Boundary Value Problems for ODEs 2013 1 / 14 Outline 1 Boundary Value
More informationAMATH 351 Mar 15, 2013 FINAL REVIEW. Instructor: Jiri Najemnik
AMATH 351 Mar 15, 013 FINAL REVIEW Instructor: Jiri Najemni ABOUT GRADES Scores I have so far will be posted on the website today sorted by the student number HW4 & Exam will be added early next wee Let
More informationAbsolute Convergence and the Ratio Test
Absolute Convergence and the Ratio Test MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Bacground Remar: All previously covered tests for convergence/divergence apply only
More information3.5 Undetermined Coefficients
3.5. UNDETERMINED COEFFICIENTS 153 11. t 2 y + ty + 4y = 0, y(1) = 3, y (1) = 4 12. t 2 y 4ty + 6y = 0, y(0) = 1, y (0) = 1 3.5 Undetermined Coefficients In this section and the next we consider the nonhomogeneous
More informationLast Update: March 1 2, 201 0
M ath 2 0 1 E S 1 W inter 2 0 1 0 Last Update: March 1 2, 201 0 S eries S olutions of Differential Equations Disclaimer: This lecture note tries to provide an alternative approach to the material in Sections
More informationLinear Variable coefficient equations (Sect. 2.1) Review: Linear constant coefficient equations
Linear Variable coefficient equations (Sect. 2.1) Review: Linear constant coefficient equations. The Initial Value Problem. Linear variable coefficients equations. The Bernoulli equation: A nonlinear equation.
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 informationVANDERBILT UNIVERSITY. MATH 2610 ORDINARY DIFFERENTIAL EQUATIONS Practice for test 1 solutions
VANDERBILT UNIVERSITY MATH 2610 ORDINARY DIFFERENTIAL EQUATIONS Practice for test 1 solutions The first test will cover all material discussed up to (including) section 4.5. Important: The solutions below
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 informationNonhomogeneous Equations and Variation of Parameters
Nonhomogeneous Equations Variation of Parameters June 17, 2016 1 Nonhomogeneous Equations 1.1 Review of First Order Equations If we look at a first order homogeneous constant coefficient ordinary differential
More informationInterpolation and the Lagrange Polynomial
Interpolation and the Lagrange Polynomial MATH 375 J. Robert Buchanan Department of Mathematics Fall 2013 Introduction We often choose polynomials to approximate other classes of functions. Theorem (Weierstrass
More informationµ = e R p(t)dt where C is an arbitrary constant. In the presence of an initial value condition
MATH 3860 REVIEW FOR FINAL EXAM The final exam will be comprehensive. It will cover materials from the following sections: 1.1-1.3; 2.1-2.2;2.4-2.6;3.1-3.7; 4.1-4.3;6.1-6.6; 7.1; 7.4-7.6; 7.8. The following
More informationMA 266 Review Topics - Exam # 2 (updated)
MA 66 Reiew Topics - Exam # updated Spring First Order Differential Equations Separable, st Order Linear, Homogeneous, Exact Second Order Linear Homogeneous with Equations Constant Coefficients The differential
More informationHigher Order Linear Equations Lecture 7
Higher Order Linear Equations Lecture 7 Dibyajyoti Deb 7.1. Outline of Lecture General Theory of nth Order Linear Equations. Homogeneous Equations with Constant Coefficients. 7.2. General Theory of nth
More informationPower Series Solutions We use power series to solve second order differential equations
Objectives Power Series Solutions We use power series to solve second order differential equations We use power series expansions to find solutions to second order, linear, variable coefficient equations
More informationProperties of Linear Transformations from R n to R m
Properties of Linear Transformations from R n to R m MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Topic Overview Relationship between the properties of a matrix transformation
More informationEquations 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 informationy + 3y = 0, y(0) = 2, y (0) = 3
MATH 3 HOMEWORK #3 PART A SOLUTIONS Problem 311 Find the solution of the given initial value problem Sketch the graph of the solution and describe its behavior as t increases y + 3y 0, y(0), y (0) 3 Solution
More informationLinear Variable coefficient equations (Sect. 1.2) Review: Linear constant coefficient equations
Linear Variable coefficient equations (Sect. 1.2) Review: Linear constant coefficient equations. The Initial Value Problem. Linear variable coefficients equations. The Bernoulli equation: A nonlinear equation.
More informationLecture 21 Power Series Method at Singular Points Frobenius Theory
Lecture 1 Power Series Method at Singular Points Frobenius Theory 10/8/011 Review. The usual power series method, that is setting y = a n 0 ) n, breaks down if 0 is a singular point. Here breaks down means
More informationUC Berkeley Math 10B, Spring 2015: Midterm 2 Prof. Sturmfels, April 9, SOLUTIONS
UC Berkeley Math 10B, Spring 2015: Midterm 2 Prof. Sturmfels, April 9, SOLUTIONS 1. (5 points) You are a pollster for the 2016 presidential elections. You ask 0 random people whether they would vote for
More informationA First Course in Elementary Differential Equations: Problems and Solutions. Marcel B. Finan Arkansas Tech University c All Rights Reserved
A First Course in Elementary Differential Equations: Problems and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 4 The Method of Variation of Parameters Problem 4.1 Solve y
More informationWorksheet # 2: Higher Order Linear ODEs (SOLUTIONS)
Name: November 8, 011 Worksheet # : Higher Order Linear ODEs (SOLUTIONS) 1. A set of n-functions f 1, f,..., f n are linearly independent on an interval I if the only way that c 1 f 1 (t) + c f (t) +...
More informationMath K (24564) - Lectures 02
Math 39100 K (24564) - Lectures 02 Ethan Akin Office: NAC 6/287 Phone: 650-5136 Email: ethanakin@earthlink.net Spring, 2018 Contents Second Order Linear Equations, B & D Chapter 4 Second Order Linear Homogeneous
More information20D - Homework Assignment 4
Brian Bowers (TA for Hui Sun) MATH 0D Homework Assignment November, 03 0D - Homework Assignment First, I will give a brief overview of how to use variation of parameters. () Ensure that the differential
More informationLinear Homogeneous ODEs of the Second Order with Constant Coefficients. Reduction of Order
Linear Homogeneous ODEs of the Second Order with Constant Coefficients. Reduction of Order October 2 6, 2017 Second Order ODEs (cont.) Consider where a, b, and c are real numbers ay +by +cy = 0, (1) Let
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 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 informationMath 1B Final Exam, Solution. Prof. Mina Aganagic Lecture 2, Spring (6 points) Use substitution and integration by parts to find:
Math B Final Eam, Solution Prof. Mina Aganagic Lecture 2, Spring 20 The eam is closed book, apart from a sheet of notes 8. Calculators are not allowed. It is your responsibility to write your answers clearly..
More information1 Solution to Homework 4
Solution to Homework Section. 5. The characteristic equation is r r + = (r )(r ) = 0 r = or r =. y(t) = c e t + c e t y = c e t + c e t. y(0) =, y (0) = c + c =, c + c = c =, c =. To find the maximum value
More informationSection 2.1 (First Order) Linear DEs; Method of Integrating Factors. General first order linear DEs Standard Form; y'(t) + p(t) y = g(t)
Section 2.1 (First Order) Linear DEs; Method of Integrating Factors Key Terms/Ideas: General first order linear DEs Standard Form; y'(t) + p(t) y = g(t) Integrating factor; a function μ(t) that transforms
More informationCalculus and Parametric Equations
Calculus and Parametric Equations MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Given a pair a parametric equations x = f (t) y = g(t) for a t b we know how
More informationMATH 251 Examination I October 8, 2015 FORM A. Name: Student Number: Section:
MATH 251 Examination I October 8, 2015 FORM A Name: Student Number: Section: This exam has 14 questions for a total of 100 points. Show all you your work! In order to obtain full credit for partial credit
More informationWork sheet / Things to know. Chapter 3
MATH 251 Work sheet / Things to know 1. Second order linear differential equation Standard form: Chapter 3 What makes it homogeneous? We will, for the most part, work with equations with constant coefficients
More informationLinear Second Order ODEs
Chapter 3 Linear Second Order ODEs In this chapter we study ODEs of the form (3.1) y + p(t)y + q(t)y = f(t), where p, q, and f are given functions. Since there are two derivatives, we might expect that
More informationChapter 5.3: Series solution near an ordinary point
Chapter 5.3: Series solution near an ordinary point We continue to study ODE s with polynomial coefficients of the form: P (x)y + Q(x)y + R(x)y = 0. Recall that x 0 is an ordinary point if P (x 0 ) 0.
More informationRank and Nullity. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics
Rank and Nullity MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Objectives We have defined and studied the important vector spaces associated with matrices (row space,
More informationHW2 Solutions. MATH 20D Fall 2013 Prof: Sun Hui TA: Zezhou Zhang (David) October 14, Checklist: Section 2.6: 1, 3, 6, 8, 10, 15, [20, 22]
HW2 Solutions MATH 20D Fall 2013 Prof: Sun Hui TA: Zezhou Zhang (David) October 14, 2013 Checklist: Section 2.6: 1, 3, 6, 8, 10, 15, [20, 22] Section 3.1: 1, 2, 3, 9, 16, 18, 20, 23 Section 3.2: 1, 2,
More information8 - Series Solutions of Differential Equations
8 - Series Solutions of Differential Equations 8.2 Power Series and Analytic Functions Homework: p. 434-436 # ü Introduction Our earlier technques allowed us to write our solutions in terms of elementary
More information