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) oscillations
and 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 1-6 (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 solution
Covers 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
Author(s): Luis Manuel Braga da Costa Campos
Series: Mathematics and Physics for Science and Technology
Publisher: CRC Press
Year: 2020
Language: English
Pages: xxxviii+222
Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
List of Classifications and Tables
Preface
Acknowledgments
About the Author
Physical Quantities
10. Examples 10.1 to 10.20
EXAMPLE 10.1: One Finite Difference (Two Differential) Equation(s) with the Same Characteristic Polynomial
E10.1.1. Linear Differential Equation with Constant Coefficients
E10.1.2. Linear Differential Equation with Homogeneous Coefficients
E10.1.3. Linear Finite Difference Equation with Constant Coefficients
EXAMPLE 10.2: Energies, Dissipation, and Power of an Oscillation
E10.2.1. Potential, Kinetic, and Total Energies
E10.2.2. Strong Subcritical or Weakly Damped Oscillations
E10.2.3. Averages over a Period of Energies and Power
E10.2.4. Balance of Forcing and Fast Amplification
EXAMPLE 10.3: Power-Law Forcing of a Harmonic Oscillator
E10.3.1. Power-Law Monotonic and Oscillating Forcing
E10.3.2. Forcing of an Undamped and Damped Oscillator
EXAMPLE 10.4: Five Standards of First-Order Differential Equations
EXAMPLE 10.5: Integrating Factor for the Linear First-Order Differential Equation
EXAMPLE 10.6: First-Order Differentials in Two Variables
E10.6.1. Exact First-Order Differentials in Two Variables
E10.6.2. Inexact First-Order Differentials in Two Variables
E10.6.3. Integrating Factors for Inexact Differentials
EXAMPLE 10.7: Paths Near a Stagnation Point of the Second Degree
E10.7.1. Paths Tangent at a Stagnation Point of the Second Degree
E10.7.2. Paths with Three Asymptotes Crossing at a Stagnation Point
E10.7.3. Comparison of Velocity Fields with a Stagnation Point
EXAMPLE 10.8: First-Order Special Differential Equations
E10.8.1. Cusped Parabola as the Envelope of a Family of Straight Lines
E10.8.2. Smooth Parabola as the Envelope of a Family of Straight Lines
E10.8.3. First-Order Equation of Degree Three without Singular Integrals
EXAMPLE 10.9: Differential Equations of Order Higher than the First
E10.9.1. Linear Differential Equations with Constant or Variable Coefficients
E10.9.2. Non-Linear Second-Order Differential Equations Omitting the Independent Variable
E10.9.3. Higher-Order Differential Equations Reducible to the First Order
E10.9.4. Linear Non-Commutative Differential Operators with Variable Coefficients
E10.9.5. Non-Linear Exact Differential Equation with Variable Coefficients
E10.9.6. Second-Order Homogeneous Differential Equation
EXAMPLE 10.10: Bending of a Beam under Traction
E10.10.1. Elastica of a Beam under Traction
E10.10.2. Clamped Beam with a Concentrated Torque
E10.10.3. Pinned Beam with a Concentrated Force
E10.10.4. Heavy Clamped-Pinned Beam
E10.10.5. Cantilever Beam with a Linearly Increasing Shear Stress
EXAMPLE 10.11: Linear Bending of a Circular Heavy Plate
with a Circular Hole
EXAMPLE 10.12: Vibrations of a Membrane under Uniform or Non-Uniform Tension
E10.12.1. Propagating/Evanescent or Cut-on/Cut-off Modes
E10.12.2. Turning Point Due to Non-Uniform Tension
EXAMPLE 10.13: Curve as the Tangent to a Vector Field or as of the Intersection Surfaces
EXAMPLE 10.14: Differentials of First Degree in Three/Four Variables
E10.14.1. Exact, Inexact, and Non-Integrable Differentials
E10.14.2. Immediate and Complete Integrability and Subsidiary Conditions
EXAMPLE 10.15: General Boundary Conditions for the Bending of a Plate
E10.15.1. Elastic Energy along the Boundary of a Plate
E10.15.2. Integration by Parts along a Closed Regular Boundary
E10.15.3. Stress Couple and Turning Moment
E10.15.4. Plate with Arbitrary Closed Regular Boundary
E10.15.5. Isotropic and Pseudo-Isotropic Orthotropic Plates
E10.15.6. General or Rectangular Clamped Plate
E10.15.7. Boundary Conditions for Supported or Pinned Plates
E10.15.8. General, Circular, or Rectangular Plate with Free Boundary
E10.15.9. Sets of Boundary Conditions for the Bending of a Plate
EXAMPLE 10.16: Strong Bending of a Pseudo-Isotropic Orthotropic Plate
EXAMPLE 10.17: Non-Linear and Linear Coupled Systems of Differential Equations
E10.17.1. Simultaneous System of Non-Linear Ordinary Differential Equations
E10.17.2. Linear Differential System with a Given Characteristic Polynomial
EXAMPLE 10.18: Three Coupled Systems with a Matrix of Characteristic Polynomials
E10.18.1. Forced Coupled Linear Differential System with Constant Coefficients
E10.18.2. Forced Coupled Linear Differential System with Homogeneous Coefficients
E10.18.3. Forced Coupled System of Finite Difference Equations
E10.18.4. System of Finite Difference Equations with Oscillatory Forcing
E10.18.5. Forcing by Power Multiplied by a Cosine
E10.18.6. Special Case of Constant Forcing
E10.18.7. General and Complete Solutions
EXAMPLE 10.19: Asymptotic Stability of a Damped Non-Linear Oscillator
E10.19.1. Positive-Definite Stability Function
E10.19.2. Negative-Definite Time Derivative of the Stability Function
EXAMPLE 10.20: Solutions of the Generalized Circular and Hyperbolic Differential Equation
E10.20.1. Simple Pole and Regular Integrals
E10.20.2. Double Pole and Elementary Integrals
E10.20.3. Triple Pole and Asymptotic Integrals
E10.20.4. High-Order Poles and Asymptotic Analytic Integrals
E10.20.5. Circular, Hyperbolic, and Airy Functions of the Inverse Variable
Conclusion 10
Classification 10.1
Classification 10.2
References
Index
