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UD351_04 - c 2004 David Russell Luke MATH351 Engineering...

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c 2004 David Russell Luke MATH351: Engineering Mathematics I (Autumn 2004) D. Russell Luke Lecture notes to follow Advanced Engineering Mathematics, 2nd Ed. by Michael Greenberg University of Delaware Autumn 2004 Department: Mathematics
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TABLE OF CONTENTS List of Figures 2 List of Tables 3 Chapter 1: Introduction to Di ff erential Equations 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Chapter 2: First Order Di ff erential Equations 6 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Examples and Applications . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Exact equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Chapter 3: Second Order Di ff erential Equations 14 3.1 Linear Dependence and Linear Independence . . . . . . . . . . . . . . 14 3.2 n -th Order Linear Homogeneous Equations: the general solution . . . 15 3.3 Solution of the Homogeneous Equation: constant coe cients . . . . . 17 3.4 A Scandalously Terse Treatment of the Harmonic Oscillator . . . . . 20 3.5 Solution of the n -th Order Inhomogeneous Equation with constant co- e cients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1
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LIST OF FIGURES 2
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LIST OF TABLES 3
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1 Chapter 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 1.1 Introduction 1. Newton’s second law of motion: acceleration = Force/mass m d 2 x dt 2 = F ( t ) where x ( t ) = spatial location and F ( t ) = force . (1.1) Integrate once to get: 2. Newton’s First Law of Motion: “An object in motion tends to stay in motion unless acted on by an outside force” (inertia). m dx dt = F 0 t + A. (1.2) Integrate once to get: 3. Ballistic path mx = 1 2 F 0 t 2 + At + B. (1.3) It’s not always so easy to solve di ff -eq’s: Example 1.1.1 m d 2 x dt 2 = kx ( t ) + F ( t ) . Formally, integration yields m dx dt + k x ( t ) dt = F ( t ) dt + A.
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2 1.2 Definitions ODE’s and PDE’s Systems of DE’s Operator notation Order Solution (on a domain) existence uniqueness (non)linear di ff eq’s (non)homogeneous di ff eq’s Initial- and boundary-value problems Definition 1.2.1 Ordinary Di ff erential Equations : “one dimensional” di ff eren- tial equations. Example 1.2.2 (ODE’s) 1. m=mass, x=displacement, k=spring sti ff ness, F=applied force: m d 2 x dt 2 + kx = F ( t ) , (Mass/Spring with restoring force) . (1.4) 2. i=current, L=inductance, C=capacitance, E=applied voltage: L d 2 i dt 2 + 1 C i = dE dt , (Electrical Circuit) . (1.5) 3. θ =angular motion, l=pendulum length, g=gravity: d 2 θ dt 2 + g l sin θ = 0 , (Pendulum) . (1.6)
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3 4. x=population, c=birth/death rate: d 2 x dt 2 = cx, (Population growth) . (1.7) 5. y=deflection, C=mass density constant: d 2 y dx 2 = C 1 + d 2 y dx 2 , (String equation) . (1.8) 6. y=deflection, w=mass loading, E&I=beam material constants: EI d 4 y dx 4 = w ( x ) , (Beam equation) . (1.9) 7. Systems of ODE’s: dx dt = ( a + b ) x + cy + dz dy dt = ax ( c + e ) y + fz (1.10) dx dt = gx + ey ( d + h ) z. describes the reactions of 3 di ff erent chemical compounds with di ff erent rate constants a, b, c, d, e, f, g, h . Definition 1.2.3 Partial Di ff erential Equations : “multi dimensional” di ff eren- tial equations. Example 1.2.4 (PDE’s) 1. u ( x, t ) = time varying heat distribution, α = di ff usivity: α 2 2 u x 2 = u t , (Heat equation) . (1.11) 2. u ( x, y, z ) = steady-state temperature distribution: 2 u x 2 + 2 u y 2 + 2 u z 2 = u = · u = 0 , (Laplace’s Equation) . (1.12)
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4 3. u ( x, y, t ) = vibrating membrane: c 2 2 u
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