Applied differential equations:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Englewood Cliffs, N.J.
Prentice-Hall
1981
|
| Edition: | 3. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | Getr. Zählung graph. Darst. |
| ISBN: | 0130400971 |
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| adam_text | contents
PREFACE xiii
= pft =
ordinary differential equations 1
CHAPTER ONE
DIFFERENTIAL EQUATIONS IN GENERAL 2
1. Concepts of Differential Equations 3
1.1 Some Definitions and Remarks 3
1.2 Simple Examples of Initial and Boundary Value Problems 7
1.3 General and Particular Solutions 15
1.4 Singular Solutions 20
? 2. Further Remarks Concerning Solutions 23
2.1 Remarks on Existence and Uniqueness 23
2.2 Direction Fields and the Method of Isoclines 28
CHAPTER TWO
FIRST ORDER AND SIMPLE HIGHER ORDER ORDINARY
DIFFERENTIAL EQUATIONS 34
1. The Method of Separation of Variables 35
2. The Method of Transformation of Variables 38
2.1 The Homogeneous Equation 38
2.2 Other Special Transformations 39
3. The Intuitive Idea of Exactness 41
4. Exact Differential Equations 43
5. Equations Made Exact by a Suitable Integrating Factor 48
v
5.1 Equations Made Exact by Integrating Factors Involving One Variable 49
5.2 The Linear First order Equation 53
5.3 The Method of Inspection 56
6. Equations of Order Higher Than the First Which are Easily Solved 57
6.1 Equations Immediately Integrable 58
6.2 Equations Having One Variable Missing 58
? 7. The Clairaut Equation 60
8. Review of Important Methods 64
CHAPTER THREE
APPLICATIONS OF FIRST ORDER AND SIMPLE HIGHER ORDER
DIFFERENTIAL EQUATIONS 70
1. Applications to Mechanics 71
1.1 Introduction 71
1.2 Newton s Laws of Motion 71
2. Applications to Electric Circuits 82
2.1 Introduction 82
2.2 Units 83
2.3 Kirchhoffs Law 84
3. Orthogonal Trajectories and Their Applications 89
4. Applications to Chemistry and Chemical Mixtures 95
5. Applications to Steady state Heat Flow 101
6. Applications to Miscellaneous Problems of Growth and Decay 106
7. The Hanging Cable 111
8. A Trip to the Moon 116
9. Applications to Rockets 120
10. Physical Problems Involving Geometry 123
11. Miscellaneous Problems in Geometry 132
12. The Deflection of Beams 137
13. Applications to Biology 148
13.1 Biological Growth 148
13.2 A Problem in Epidemiology 153
13.3 Absorption of Drugs in Organs or Cells 156
14. Applications to Economics 159
14.1 Supply and Demand 159
14.2 Inventory 162
CHAPTER FOUR
LINEAR DIFFERENTIAL EQUATIONS 166
1. The General nth order Linear Differential Equation 167
2. Existence and Uniqueness of Solutions of Linear Equations 171
3. How Do We Obtain the Complementary Solution? 173
3.1 The Auxilary Equation 173
3.2 The Case of Repeated Roots 175 ;
3.3 The Case of Imaginary Roots 178
3.4 Linear Independence and Wronskians 181
vi
» 4. How Do We Obtain a Particular Solution? 191
4.1 Method of Undetermined Coefficients 191
4.2 Justification for the Method of Undetermined Coefficients.
The Annihilator Method 193
4.3 Exceptions in the Method of Undetermined Coefficients 195
4.4 Cases Where More Complicated Functions Appear on the Right hand Side 198
4.5 The Method of Variation of Parameters 200
4.6 Short cut Methods Involving Operators 204
5. Remarks Concerning Equations with Variable Coefficients
Which Can Be Transformed into Linear Equations
with Constant Coefficients: The Euler Equation 211
6. Review of Important Methods 214
CHAPTER FIVE
APPLICATIONS OF LINEAR DIFFERENTIAL EQUATIONS 219
1. Vibratory Motion of Mechanical Systems 220
1.1 The Vibrating Spring. Simple Harmonic Motion 220
1.2 The Vibrating Spring with Damping. Overdamped
and Critically Damped Motion 228
1.3 The Spring with External Forces 236
1.4 The Phenomenon of Mechanical Resonance 238
2. Electric Circuit Problems 242
3. Miscellaneous Problems 246
3.1 The Simple Pendulum 246
3.2 Vertical Oscillations of a Box Floating in a Liquid 247
3.3 A Problem in Cardiography 249
3.4 An Application to Economics 250
CHAPTER SIX
SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS
BY LAPLACE TRANSFORMS 255
1. Introduction to the Method of Laplace Transforms 256
1.1 Motivation for Laplace Transforms 256
1.2 Definition and Examples of the Laplace Transform 256
1.3 Further Properties of Laplace Transforms 260
1.4 The Gamma Function 261
1.5 Remarks Concerning Existence of Laplace Transforms 262
1.6 The Heaviside Unit Step Function 264
2. Impulse Functions and the Dirac Delta Function 268
3. Application of Laplace Transforms to Differential Equations 273
3.1 Solution of Simple Differential Equations. Inverse Laplace Transforms 273
3.2 Some Methods for Finding Inverse Laplace Transforms 274
3.3 Remarks Concerning Existence and Uniqueness
of Inverse Laplace Transforms 282
4. Applications to Physical and Biological Problems 285
4.1 Applications to Electric Circuits 285
vii
4.2 An Application to Biology 288
4.3 The Tautochrone Problem—An Integral Equation Application to Mechanics 289
4.4 Applications Involving the Delta Function 293
4.5 An Application to the Theory of Automatic Control and Servomechanisms 294
CHAPTER SEVEN
SOLUTION OF DIFFERENTIAL EQUATIONS BY USE
OF SERIES 299
1. Introduction to the Use of Series 300
1.1 Motivation for Series Solutions 300
1.2 Use of the Summation Notation 302
1.3 Some Questions of Rigor 306
1.4 The Taylor Series Method 312
1.5 Picard s Method of Iteration 314
2. The Method of Frobenius 317
2.1 Motivation for the Method of Frobenius 317
2.2 Examples Using the Method of Frobenius 321
3. Series Solutions of Some Important Differential Equations 333
3.1 Bessel s Differential Equation 333
3.2 Legendre s Differential Equation 343
3.3 Other Special Functions 345
CHAPTER EIGHT
^ ORTHOGONAL FUNCTIONS AND STURM LIOUVILLE
PROBLEMS 348
1. Orthogonal Functions 349
1.1 Functions as Vectors 349
1.2 Orthogonality 351
1.3 Length or Norm of a Vector. Orthonormality 352
2. Sturm Liouville Problems 356
2.1 Motivation for Sturm Liouville Problems. Eigenvalues and Eigenfunctions 356
2.2 An Application to the Buckling of Beams 363
3. Orthogonality of Bessel and Legendre Functions 366
3.1 Orthogonality of Bessel Functions 366
3.2 Orthogonality of Legendre Functions 371
3.3 Miscellaneous Orthogonal Functions 373
4. Orthogonal Series 375
4.1 Introduction 375
4.2 Fourier Series 380
4.3 Bessel Series 397
4.4 Legendre Series 402
4.5 Miscellaneous Orthogonal Series 405
5. Some Special Topics 407 *
5.1 Self adjoint Differential Equations 407
5.2 The Gram Schmidt Orthonormalization Method 410
viii
CHAPTER NINE
THE NUMERICAL SOLUTION OF DIFFERENTIAL
EQUATIONS 414
1. Numerical Solution of y =f(x,y) 415
1.1 The Constant Slope or Euler Method 416
1.2 The Average Slope or Modified Euler Method 419
1.3 Computer Diagrams 421
1.4 Error Analysis 422
1.5 Some Practical Guidelines for Numerical Solution 425
2. The Runge Kutta Method 427
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systems of ordinary differential equations
CHAPTER TEN
SYSTEMS OF DIFFERENTIAL EQUATIONS
AND THEIR APPLICATIONS 432
1. Systems of Differential Equations 433
1.1 Motivation for Systems of Differential Equations 433
1.2 Method of Elimination for Solving Systems of Differential Equations 434
1.3 The Use of Operators in Eliminating Unknowns 436
1.4 Short cut Operator Methods 440
2. Solutions of Non linear Systems of Ordinary Differential Equations 441
3. Differential Equations Expressed as First order Systems 443
4. Application to Mechanics 446
4.1 The Flight of a Projectile 446
4.2 An Application to Astronomy 451
4.3 The Motion of Satellites and Missiles 458
4.4 The Problem of the Vibrating Masses 463
5. Applications to Electric Networks 469
6. Applications to Biology 473
6.1 Concentration of a Drug in a Two compartment System 473
6.2 The Problem of Epidemics with Quarantine 476
7. The Predator Prey Problem: A Problem in Ecology 480
7.1 Mathematical Formulation 481
7.2 Investigation of a Solution 481
7.3 Some Further Applications 488
8. Solutions of Linear Systems by Laplace Transforms 489
9. Method of Complementary and Particular Solutions 491
ix
9.1 How Do We Find the Complementary Solution? 493
9.2 How Do We Find a Particular Solution? 497
9.3 Summary of Procedure 498
CHAPTER ELEVEN
^ MATRIX EIGENVALUE METHODS FOR SYSTEMS OF LINEAR
DIFFERENTIAL EQUATIONS 501
1. The Concept of a Matrix 502
1.1 Introduction 502
1.2 Some Simple Ideas 502
1.3 Row and Column Vectors 503
1.4 Operations with Matrices 505
2. Matrix Differential Equations 512
3. The Complementary Solution 513
3.1 Eigenvalues and Eigenvectors 514
3.2 The Case of Real Distinct Eigenvalues 515
3.3 The Case of Repeated Eigenvalues 517
3.4 The Case of Imaginary Eigenvalues 518
3.5 A Slightly More Complicated Problem 520
3.6 Linear Independence and Wronskians 522
4. The Particular Solution 524
5. Summary of Procedure 525
6. Application Using Matrices 526
7. Some Special Topics 530
7.1 Orthogonality 530
7.2 Length of a Vector 531
7.3 Eigenvalues and Eigenvectors of Real Symmetric Matrices 533
partial differential equations
CHAPTER TWELVE
PARTIAL DIFFERENTIAL EQUATIONS IN GENERAL 540
1. The Concept of a Partial Differential Equation 541
1.1 Introduction 541
1.2 Solutions of Some Simple Partial Differential Equations 541
1.3 Geometric Significance of General and Particular Solutions 544
1.4 Partial Differential Equations Arising from Elimination of Arbitrary Functions 545
2. The Method of Separation of Variables 550
3. Some Important Partial Differential Equations Arising
from Physical Problems 558
x
3.1 Problems Involving Vibrations or Oscillations: The Vibrating String 558
3.2 Problems Involving Heat Conduction or Diffusion 562
3.3 Problems Involving Electrical or Gravitational Potential 566
3.4 Remarks on the Derivation of Partial Differential Equations 567
CHAPTER THIRTEEN
SOLUTIONS OF BOUNDARY VALUE PROBLEMS
USING FOURIER SERIES 570
1. Boundary Value Problems Involving Heat Conduction 571
1.1 Fourier s Problem 571
1.2 Problems Involving Insulated Boundaries 577
1.3 Steady state Temperature in a Semi infinite Plate 579
1.4 Diffusion Interpretation of Heat Conduction 581
2. Boundary Value Problems Involving Laplace s Equation 585
2.1 The Vibrating String Problem 585
2.2 The Vibrating String with Damping 589
2.3 Vibrations of a Beam 591
3. Boundary Value Problems Involving Laplace Equation 595
4. Miscellaneous Problems 602
4.1 The Vibrating String Under Gravity 603
4.2 Heat Conduction in a Bar with Non zero End Conditions 604
4.3 The Vibrating String with Non zero Initial Velocity 606
4.4 Vibrations of a Square Drumhead: A Problem Involving
Double Fourier Series 608
4.5 Heat Conduction with Radiation 612
CHAPTER FOURTEEN
^ SOLUTIONS OF BOUNDARY VALUE PROBLEMS USING BESSEL
AND LEGENDRE FUNCTIONS 619
1. Introduction 620
2. Boundary Value Problems Leading to Bessel Functions 620
2.1 The Laplacian in Cylindrical Coordinates 620
2.2 Heat Conduction in a Circular Cylinder 621
2.3 Heat Conduction in a Radiating Cylinder 624
2.4 Vibrations of a Circular Drumhead 625
3. Boundary Value Problems Leading to Legendre Functions 633
3.1 The Laplacian in Spherical Coordinates 633
3.2 Heat Conduction in a Sphere 634
3.3 Electrical or Gravitational Potential Due to a Sphere 637
4. Miscellaneous Problems 641
4.1 The Problem of the Vibrating Chain 641
4.2 Electrical Potential Due to Uniformly Charged Circular Wire 645
4.3 The Atomic Bomb Problem 648
xi
APPENDIX
DETERMINANTS A 1
ANSWERS TO EXERCISES A 7
BIBLIOGRAPHY B 1
INDEX 1 1
xii
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| isbn | 0130400971 |
| language | English |
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| spelling | Spiegel, Murray R. 1923-1991 Verfasser (DE-588)109314840 aut Applied differential equations 3. ed. Englewood Cliffs, N.J. Prentice-Hall 1981 Getr. Zählung graph. Darst. txt rdacontent n rdamedia nc rdacarrier Équations différentielles Differential equations Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003960060&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Spiegel, Murray R. 1923-1991 Applied differential equations Équations différentielles Differential equations Differentialgleichung (DE-588)4012249-9 gnd |
| subject_GND | (DE-588)4012249-9 |
| title | Applied differential equations |
| title_auth | Applied differential equations |
| title_exact_search | Applied differential equations |
| title_full | Applied differential equations |
| title_fullStr | Applied differential equations |
| title_full_unstemmed | Applied differential equations |
| title_short | Applied differential equations |
| title_sort | applied differential equations |
| topic | Équations différentielles Differential equations Differentialgleichung (DE-588)4012249-9 gnd |
| topic_facet | Équations différentielles Differential equations Differentialgleichung |
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