notes152-ch3

# notes152-ch3 - MATH 152 Spring 2004-05 Applied Linear...

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Unformatted text preview: MATH 152 Spring 2004-05 Applied Linear Algebra & Differential Equations Lecture Notes Dr. Tony Yee Department of Mathematics The Hong Kong University of Science and Technology February 15, 2005 ii Contents Table of Contents iii 1 Introduction 3 1.1 What are Differential Equations? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Classification of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Solutions of Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 5 2 First-Order Differential Equations 11 2.1 First-Order Linear Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 The Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 Modeling with First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.6 Euler and Runge–Kutta Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . 35 3 Second-Order Linear Equations 43 3.1 Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.1.1 Homogeneous Equations with Real Constant Coefficients . . . . . . . . . . . 46 3.1.2 Method of Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 Nonhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2.1 Method of Undetermined Coefficients . . . . . . . . . . . . . . . . . . . . . . 55 3.2.2 Method of Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3 Mechanical Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 iii Chapter 3 Second-Order Linear Equations Many engineering problems can yield second-order linear differential equations. For example, consider the motion of an object of mass m attached to a spring with spring constant k . Let y ( t ) denote the displacement at time t of the mass from its equilibrium position. Combining Newton’s Second Law of Motion and Hook’s Law, the model equation will be force acting on the object = m d 2 y dt 2 ( t ) =- ky ( t ) , or simply y 00 + k m y = 0 . This model equation is called a simple harmonic oscillator and is indeed a second-order linear equation. A second-order linear ordinary differential equation has the most general form y 00 + p ( t ) y + q ( t ) y = g ( t ) . (3.1) The key property of Eq (3.1) is its linear property. Linear Property If y 1 and y 2 are respective solutions of y 00 + p ( t ) y + q ( t ) y = g i ( t ) , i = 1 , 2 , then for any constants c 1 and c 2 , y = c 1 y 1 + c 2 y 2 is a solution of y 00 + p ( t ) y + q ( t ) y = c 1 g 1 ( t ) + c 2 g 2 ( t ) ....
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notes152-ch3 - MATH 152 Spring 2004-05 Applied Linear...

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