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HigherLin1 - HIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL...

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HIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS I: Introduction and Homogeneous Equations David Levermore Department of Mathematics University of Maryland 30 August 2009 Because the presentation of this material in class will differ somewhat from that in the book, I felt that notes that closely follow the class presentation might be appreciated. 1. Introduction 1.1. Normal Form and Solutions 2 1.2. Initial Value Problems 2 1.3. Intervals of Existence 4 2. Homogeneous Equations: General Theory 2.1. Linear Superposition 5 2.2. Wronskians 8 2.3. Fundamental Sets of Solutions and General Solutions 9 2.4. Linear Independence of Solutions 11 2.5. Reduction of Order 15 3. Homogeneous Equations with Constant Coefficients 3.1. Characteristic Polynomials and the Key Identity 18 3.2. Real Roots of Characteristic Polynomials 19 3.3. Complex Roots of Characteristic Polynomials 22 A. Linear Algebraic Systems and Determinants A.1. Linear Algebraic Systems 27 A.2. Determinants 28 A.3. Existence of Solutions 30 1
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2 1. Introdution 1.1: Normal Form and Solutions. An n th -order linear ordinary differential equation can be brought into the linear normal form d n y dt n + a 1 ( t ) d n 1 y dt n 1 + · · · + a n 1 ( t ) dy dt + a n ( t ) y = f ( t ) . (1.1) Here a 1 ( t ) , · · · , a n ( t ) are call coefficients while f ( t ) is called the forcing or driving . When f ( t ) = 0 the equation is said to be homogeneous ; otherwise it is said to be nonhomogeneous . Definition: We say that y = Y ( t ) is a solution of (1.1) over an interval ( t L , t R ) provided that: the function Y is n -times differentiable over ( t L , t R ), the coefficients a 1 ( t ), a 2 ( t ), · · · , a n ( t ), and the forcing f ( t ) are defined over ( t L , t R ), the equation Y ( n ) ( t ) + a 1 ( t ) Y ( n 1) ( t ) + · · · + a n 1 ( t ) Y ( t ) + a n ( t ) Y ( t ) = f ( t ) is satisfied for every t in ( t L , t R ). The first two bullets simply say that every term appearing in the equation is defined over the interval ( t L , t R ), while the third says the equation is satisfied at each time t in ( t L , t R ). 1.2: Initial-Value Problem. An initial-value problem associated with (1.1) seeks a solution y = Y ( t ) of (1.1) that also satisfies the initial conditions Y ( t I ) = y 0 , Y ( t I ) = y 1 , · · · Y ( n 1) ( t I ) = y n 1 , (1.2) for some initial time (or initial point ) t I and initial data (or initial values ) y 0 , y 1 , · · · , y n 1 . You should know the following basic existence and uniqueness theorem about initial-value problems, which we state without proof. Theorem 1.1 (Basic Existence and Uniqueness Theorem): Let the func- tions a 1 , a 2 , · · · , a n , and f all be continuous over an interval ( t L , t R ). Then given any initial time t I ( t L , t R ) and any initial data y 0 , y 1 , · · · , y n 1 there exists a unique solution y = Y ( t ) of (1.1) that satisfies the initial conditions (1.2). Moreover, this solution has at least n continuous derivatives over ( t L , t R ). If the functions a 1 , a 2 , · · · , a n , and f all have k continuous derivatives over ( t L , t R ) then this solution has at least k + n continuous derivatives over ( t L , t R ).
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