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Unformatted text preview: ORDINARY DIFFERENTIAL EQUATION: Introduction and FirstOrder Equations David Levermore Department of Mathematics University of Maryland 7 September 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, Classification, and Overview 1.1. Introduction 2 1.2. Classification 3 1.3. Course Overview 4 2. FirstOrder Equations: General and Explicit 2.1. FirstOrder Equations 6 2.2. Explicit Equations 7 3. FirstOrder Equations: Linear 3.1. Linear Normal Form 10 3.2. Recipe for Homogeneous Linear Equations 11 3.3. Recipe for Nonhomogenous Linear Equations 12 3.4. Linear InitialValue Problems 14 3.5. Theory for Linear Equations 15 4. FirstOrder Equations: Separable 4.1. Recipe for Autonomous Equations 17 4.2. Recipe for Separable Equations 21 4.3. Separable InitialValue Problems 24 4.4. Theory for Separable Equations 30 5. FirstOrder Equations: Graphical Methods 5.1. PhaseLine Portraits for Autonomous Equations 32 5.2. Plots of Explicit Solutions 34 5.3. Contour Plots of Implicit Solutions 34 5.4. Direction Fields 34 1 2 1. Introduction, Classification, and Overview 1.1. Introduction. We begin with a definition illustrated by examples. Definition. A differential equation is an algebraic relation involving derivatives of one or more unknown functions with respect to one or more independent vari ables, and possibly either the unknown functions themselves or their independent variables, that hold at each point in the domain of those functions. For example, an unknown function p ( t ) might satisfy the relation dp dt = 5 p. (1.1) This is a differential equation because it algebraically relates the derivative of the unknown function p to itself. It does not involve the independent variable t . It is understood that this relation should hold every point t where p ( t ) and its derivative are defined. Similarly, unknown functions u ( x, y ) and v ( x, y ) might satisfy the relation ∂ x u + ∂ y v = 0 , (1.2) where ∂ x u and ∂ y v denote partial derivatives. This is a differential equation because it algebraically relates some partial derivatives of the unknown functions u and v to each other. It does not involve the values of either u or v , nor does it involve either of the independent variables, x or y . It is understood that this relation should hold every point ( x, y ) where u ( x, y ), v ( x, y ) and their partial derivatives appearing in (1.2) are defined. Here are other examples of differential equations that involve derivatives of a single unkown function: (a) dv dt = 9 . 8 − . 1 v 2 , (c) parenleftbigg dy dx parenrightbigg 2 + x 2 + y 2 = − 1 , (e) d 2 u dr 2 + 2 r du dr + u = 0 , (g) d 2 x dt 2 + 9 x = cos(2 t ) , (i) d 2 A dx 2 + xA = 0 , (k) ∂ tt h = ∂ xx h + ∂ yy h, (m) ∂ t u + u∂ x u = ∂ xx u , (b) d 2 θ dt 2 + sin( θ ) = 0 , (d) parenleftbigg dy dx parenrightbigg 2 + 4 y 2 = 1...
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This note was uploaded on 01/24/2011 for the course MATH math246 taught by Professor Levermore during the Spring '10 term at Maryland.
 Spring '10
 LEVERMORE

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