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Unformatted text preview: ORDINARY DIFFERENTIAL EQUATION: Introduction and First-Order 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. First-Order Equations: General and Explicit 2.1. First-Order Equations 6 2.2. Explicit Equations 7 3. First-Order 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 Initial-Value Problems 14 3.5. Theory for Linear Equations 15 4. First-Order Equations: Separable 4.1. Recipe for Autonomous Equations 17 4.2. Recipe for Separable Equations 21 4.3. Separable Initial-Value Problems 24 4.4. Theory for Separable Equations 30 5. First-Order Equations: Graphical Methods 5.1. Phase-Line 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