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E-raamat: Classification and Examples of Differential Equations and their Applications: Ordinary Differential Equations with Applications to Trajectories and Vibrations [Taylor & Francis e-raamat]

(University of Lisbon, Portugal)
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Classification and Examples of Differential Equations and their Applications: Ordinary Differential Equations with Applications to Trajectories and VibrationsSuurem pilt
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Taylor & Francis e-raamatudRohkem infot Taylor & Francis e-raamatute kohta
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Classification and Examples of Differential Equations and their Applications is the sixth book within Ordinary Differential Equations with Applications to Trajectories and Vibrations, Six-volume Set. As a set they are the fourth volume in the series Mathematics and Physics Applied to Science and Technology. This sixth book consists of one chapter (chapter 10 of the set). It contains 20 examples related to the preceding five books and chapters 1 to 9 of the set. It includes two recollections the first with a classification of differential equations into 500 standards and the second with a list of 500 applications.The ordinary differential equations are classified in 500 standards concerning methods of solution and related properties, including: (i) linear differential equations with constant or homogeneous coefficients and finite difference equations; (ii) linear and non-linear single differential equations and simultaneous systems; (iii) existence, unicity and other properties; (iv) derivation of general, particular, special, analytic, regular, irregular and normal integrals; (v) linear differential equations with variable coefficients including known and new special functions. The theory of differential equations is applied to the detailed solution of 500 physical and engineering problems including: (i) one- and multidimensional oscillators, with damping or amplification, with non-resonant or resonant forcing; (ii) single, non-linear and parametric resonance; (iii) bifurcations and chaotic dynamical systems; (iv) longitudinal and transversal deformations and buckling of bars, beams and plates; (v) trajectories of particles; (vi) oscillationsand waves in non-uniform media, ducts and wave guides.Provides detailed solution of examples of differential equations of the types covered in tomes l-5 of the set (Ordinary Differential Equations with Applications to Trajectories and Vibrations, Six -volume Set)Includes physical and engineering problems that extend those presented in the tomes 1 to 6 (Ordinary Differential Equations with Applications to Trajectories and Vibrations, Six-volume Set)Includes a classification of ordinary differential equations and their properties into 500 standards that can serve as a look-up table of methods of solutionCovers a recollection of 500 physical and engineering problems and sub-cases that involve the solution of differential equations Presents the problems used as examples including formulation, solution, and interpretation of results
Contents vii
List of Classifications and Tables
xi
Preface xxi
Acknowledgments xxv
About the Author xxvii
Physical Quantities xxix
10 Examples 10.1 to 10.20
1(91)
Example 10.1 One Finite Difference (Two Differential) Equation(s) with the Same Characteristic Polynomial
1(4)
E10.1.1 Linear Differential Equation with Constant Coefficients
1(2)
E10.1.2 Linear Differential Equation with Homogeneous Coefficients
3(1)
E10.1.3 Linear Finite Difference Equation with Constant Coefficients
4(1)
Example 10.2 Energies, Dissipation, and Power of an Oscillation
5(10)
E10.2.1 Potential, Kinetic, and Total Energies 5 E10.2.2 Strong Subcritical or Weakly Damped Oscillations
7(1)
E10.2.3 Averages over a Period of Energies and Power
8(3)
E10.2.4 Balance of Forcing and Fast Amplification 10 EXAMPLE 10.3 Power-Law Forcing of a Harmonic Oscillator
11(1)
E10.3.1 Power-Law Monotonic and Oscillating Forcing
12(1)
E10.3.2 Forcing of an Undamped and Damped Oscillator
13(2)
Example 10.4 Five Standards of First-Order Differential Equations
15(2)
Example 10.5 Integrating Factor for the Linear First-Order Differential Equation
17(1)
Example 10.6 First-Order Differentials in Two Variables
18(3)
E10.6.1 Exact First-Order Differentials in Two Variables
19(1)
E10.6.2 Inexact First-Order Differentials in Two Variables
19(1)
E10.6.3 Integrating Factors for Inexact Differentials
20(1)
Example 10.7 Paths Near a Stagnation Point of the Second Degree
21(4)
E10.7.1 Paths Tangent at a Stagnation Point of the Second Degree
21(2)
E10.7.2 Paths with Three Asymptotes Crossing at a Stagnation Point
23(1)
E10.7.3 Comparison of Velocity Fields with a Stagnation Point
24(1)
Example 10.8 First-Order Special Differential Equations
25(3)
E10.8.1 Cusped Parabola as the Envelope of a Family of Straight Lines
25(1)
E10.8.2 Smooth Parabola as the Envelope of a Family of Straight Lines
26(1)
E10.8.3 First-Order Equation of Degree Three without Singular Integrals
27(1)
Example 10.9 Differential Equations of Order Higher than the First
28(7)
E10.9.1 Linear Differential Equations with Constant or Variable Coefficients
29(1)
E10.9.2 Non-Linear Second-Order Differential Equations Omitting the Independent Variable
30(1)
E10.9.3 Higher-Order Differential Equations Reducible to the First Order
31(1)
E10.9.4 Linear Non-Commutative Differential Operators with Variable Coefficients
32(1)
E10.9.5 Non-Linear Exact Differential Equation with Variable Coefficients
33(1)
E10.9.6 Second-Order Homogeneous Differential Equation
34(1)
Example 10.10 Bending of a Beam under Traction
35(13)
E10.10.1 Elastica of a Beam under Traction
36(1)
E10.10.2 Clamped Beam with a Concentrated Torque
37(3)
E10.10.3 Pinned Beam with a Concentrated Force
40(3)
E10.10.4 Heavy Clamped-Pinned Beam
43(2)
E10.10.5 Cantilever Beam with a Linearly Increasing Shear Stress
45(3)
Example 10.11 Linear Bending of a Circular Heavy Plate with a Circular Hole
48(3)
Example 10.12 Vibrations of a Membrane under Uniform or Non-Uniform Tension
51(5)
E10.12.1 Propagating/Evanescent or Cut-on/Cut-off Modes
54(1)
E10.12.2 Turning Point Due to Non-Uniform Tension
55(1)
Example 10.13 Curve as the Tangent to a Vector Field or as of the Intersection Surfaces
56(2)
Example 10.14 Differentials of First Degree in Three/Four Variables
58(2)
E10.14.1 Exact, Inexact, and Non-Integrable Differentials
58(1)
E10.14.2 Immediate and Complete Integrability and Subsidiary Conditions
59(1)
Example 10.15 General Boundary Conditions for the Bending of a Plate
60(8)
E10.15.1 Elastic Energy along the Boundary of a Plate
61(1)
E10.15.2 Integration by Parts along a Closed Regular Boundary
62(1)
E10.15.3 Stress Couple and Turning Moment
63(1)
E10.15.4 Plate with Arbitrary Closed Regular Boundary
64(1)
E10.15.5 Isotropic and Pseudo-Isotropic Orthotropic Plates
65(1)
E10.15.6 General or Rectangular Clamped Plate
65(1)
E10.15.7 Boundary Conditions for Supported or Pinned Plates
66(1)
E10.15.8 General, Circular, or Rectangular Plate with Free Boundary
67(1)
E10.15.9 Sets of Boundary Conditions for the Bending of a Plate
67(1)
Example 10.16 Strong Bending of a Pseudo-Isotropic Orthotropic Plate
68(3)
Example 10.17 Non-Linear and Linear Coupled Systems of Differential Equations
71(3)
E10.17.1 Simultaneous System of Non-Linear Ordinary Differential Equations
71(1)
E10.17.2 Linear Differential System with a Given Characteristic Polynomial
72(2)
Example 10.18 Three Coupled Systems with a Matrix of Characteristic Polynomials
74(10)
E10.18.1 Forced Coupled Linear Differential System with Constant Coefficients
75(2)
E10.18.2 Forced Coupled Linear Differential System with Homogeneous Coefficients
77(1)
E10.18.3 Forced Coupled System of Finite Difference Equations
77(1)
E10.18.4 System of Finite Difference Equations with Oscillatory Forcing
78(1)
E10.18.5 Forcing by Power Multiplied by a Cosine
78(2)
E10.18.6 Special Case of Constant Forcing
80(1)
E10.18.7 General and Complete Solutions
81(1)
Example 10.19 Asymptotic Stability of a Damped Non-Linear Oscillator
82(1)
E10.19.1 Positive-Definite Stability Function
83(1)
E10.19.2 Negative-Definite Time Derivative of the Stability Function
83(1)
Example 10.20 Solutions of the Generalized Circular and Hyperbolic Differential Equation
84(8)
E10.20.1 Simple Pole and Regular Integrals
85(2)
E10.20.2 Double Pole and Elementary Integrals
87(1)
E10.20.3 Triple Pole and Asymptotic Integrals
88(2)
E10.20.4 High-Order Poles and Asymptotic Analytic Integrals
90(1)
E10.20.5 Circular, Hyperbolic, and Airy Functions of the Inverse Variable
91(1)
Conclusion 10 92(2)
Classification 10.1 94(65)
Classification 10.2 159(44)
References 203(6)
Index 209
Luis Manuel Braga da Costa Campos graduated in 1972 as a Mechanical Engineer from the Instituto Superior Tecnico (IST) of Lisbon Technical University. His tutorials as a student (1970) were followed by a career at the same institution (IST) through all levels: Assistant (1972), Assistant with tenure (1974), Assistant Professor (1978), Associate Professor (1982), Chair de Applied Mathematics and Mechanics (1985). He has been coordinator of undergraduate and post-graduate degrees in Aerospace Engineering since their creation in 1991. He is also coordinator of the Scientific Area of Applied and Aerospace Mechanics in the Department of Mechanical Engineering and director and founder of the Center for Aeronautical and Space Science and Technology.
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