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Unformatted text preview: Notes on Differential Equations Thomas Kriete Rebecca Schmitz Erin Valenti Copyright c circlecopyrt 2005 Thomas L. Kriete All rights reserved. Contents Unit 1 1.1 Direction Fields 1.2 Solutions of Some Differential Equations 1.3 Classification of Differential Equations 1 Unit 2 2.1 Linear Equations 2.2 Separable Equations 2.3 Modeling 8 Unit 3 2.4 Linear and Nonlinear Equations 2.5 Population Dynamics 2.6 Exact Equations; Integrating Factors 17 Unit 4 3.0 Introduction to Chapter 3 3.1 Homogenous Equations with Constant Coefficients 3.2 Fundamental Solutions of Linear Homogenous Equations 3.3 Linear Independence and the Wronskian 35 Unit 5 3.4 Complex Roots of the Characteristic Equation 3.5 Repeated Roots; Reduction of Order 3.7 Variation of Parameters 51 Unit 6 4.1 General Theory of n th Order Linear Equations 4.2 Homogeneous Equations with Constant Coefficients 3.6/4.3 Nonhomogeneous Equations and Annihilating Operators 64 Unit 7 3.8 Mechanical and Electrical Vibrations 3.9 Forced Vibrations 81 Unit 8 5.1 Review of Power Series 5.2/5.3 Series Solutions Near an Ordinary Point 93 Unit 9 7.1 Introduction i 7.2, 7.3 Review of Matrices 7.47.6, 9.1 Solving x prime = Ax via Eigenvalues and Eigenvectors 101 Unit 10 7.7, 7.8, 9.1 The Exponential Matrix e tA , and Using it to Solve the System x prime = Ax 122 Unit 11 9.2 Autonomous Systems and Stability 9.3 Almost Linear Systems 9.4 Competing Systems 9.5 PredatorPrey Equations 138 Unit 12 10.2 Fourier Series 10.3 The Fourier Convergence Theorem 10.4 Even and Odd Functions 154 Unit 13 10.1 TwoPoint Boundary Value Problems 10.5 Heat Conduction in a Rod 168 ii Unit 1 1.1 Direction Fields 1.2 Solutions of Some Differential Equations 1.3 Classification of Differential Equations Reading Assignment Chapter 1; pp. 129 Learning Objectives The student will be able to draw direction fields using isoclines classify certain types of differential equations solve certain firstorder linear differential equations verify solutions of given differential equations Commentary Section 1.1 Differential equations are just equations containing unknown functions and their derivatives. They are used to understand and solve important problems in the sciences, finance and other disciplines. A differential equation that describes some physical process is called a mathematical model of the process. In Section 1.1, you will be introduced to direction fields (also commonly known as slope fields). Direction fields give us information about the solutions of differential equations of the form dy dt = f ( t,y ) without having to solve the differential equation (note: the text (and we) sometimes use the independent variable x instead of t ). The differential equation tells you the slope of a solution at the point ( t,y ). To solve, find the curve y ( t ). In more detail, if y ( t ) is the function that solves the differential equation y prime ( t ) = f ( t,y ( t )), then the differential equation tells us the slope y prime...
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This note was uploaded on 03/25/2009 for the course MATH 325 taught by Professor Mitria during the Spring '07 term at UVA.
 Spring '07
 Mitria
 Equations

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