Elementary Differential Equations

by Rainville, Earl D.; Bedient, Phillip E.; Bedient, Richard E.

• ISBN13: 9780135080115
• ISBN10: 0135080118
• Edition: 8th
• Format: Paperback
• Copyright: 1996-10-23
• Publisher: Pearson

Summary

A clear, concise book that emphasizes finding solutions to differential equations where applications play an important role. Each chapter includes many illustrative examples to assist the reader.The book emphasizes methods for finding solutions to differential equations. It provides many abundant exercises, applications, and solved examples with careful attention given to readability.Elementary Differential Equationsincludes a thorough treatment of power series techniques. In addition, the book presents a classical treatment of several physical problems to show how Fourier series become involved in the solution of those problems. The eighth edition ofElementary Differential Equationshas been revised to include a new supplement in many chapters that provides suggestions and exercises for using a computer to assist in the understanding of the material in the chapter. It also now provides an introduction to the phase plane and to different types of phase portraits.A valuable reference book for readers interested in exploring the technological and other applications of differential equations.

Table of Contents

Preface	xiii
1 Definitions; Families of Curves	1	(17)
1.1 Examples of Differential Equations	1	(1)
1.2 Definitions	2	(3)
1.3 Families of Solutions	5	(5)
1.4 Geometric Interpretation	10	(2)
1.5 The Isoclines of an Equation	12	(2)
1.6 An Existence Theorem	14	(1)
1.7 Computer Supplement	15	(3)
2 Equations of Order One	18	(27)
2.1 Separation of Variables	18	(6)
2.2 Homogeneous Functions	24	(1)
2.3 Equations with Homogeneous Cofficients	25	(4)
2.4 Exact Equations	29	(6)
2.5 The Linear Equation of Order One	35	(3)
2.6 The General Solution of a Linear Equation	38	(5)
2.7 Computer Supplement	43	(2)
3 Numerical Methods	45	(17)
3.1 General Remarks	45	(1)
3.2 Euler's Method	45	(3)
3.3 A Modification of Euler's Method	48	(1)
3.4 A Method of Successive Approximation	49	(2)
3.5 An Improvement on the Method of Successive Approximation	51	(1)
3.6 The Use of Taylor's Theorem	52	(2)
3.7 The Runge-Kutta Method	54	(4)
3.8 A Continuing Method	58	(2)
3.9 Computer Supplement	60	(2)
Elementary Applications	62	(13)
4.1 Velocity of Escape from the Earth	62	(2)
4.2 Newton's Law of Cooling	64	(1)
4.3 Simple Chemical Conversion	65	(4)
4.4 Logistic Growth and the Price of Commodities	69	(4)
4.5 Computer Supplement	73	(2)
5 Additional Topics on Equations of Order One	75	(24)
5.1 Integrating Factors Found by Inspection	75	(4)
5.2 The Determination of Integrating Factors	79	(4)
5.3 Substitution Suggested by the Equation	83	(3)
5.4 Bernoulli's Equation	86	(3)
5.5 Coefficients Linear in the Two Variables	89	(5)
5.6 Solutions Involving Nonelementary Integrals	94	(3)
5.7 Computer Supplement	97	(2)
6 Linear Differential Equations	99	(18)
6.1 The General Linear Equation	99	(1)
6.2 An Existence and Uniqueness Theorem	100	(2)
6.3 Linear Independence	102	(1)
6.4 The Wronskian	103	(3)
6.5 General Solution of a Homogeneous Equation	106	(1)
6.6 General Solution of a Nonhomogeneous Equation	107	(2)
6.7 Differential Operators	109	(1)
6.8 The Fundamental Laws of Operation	111	(2)
6.9 Some Properties of Differential Operators	113	(2)
6.10 Computer Supplement	115	(2)
7 Linear Equations with Constant Coefficients	117	(17)
7.1 Introduction	117	(1)
7.2 The Auxiliary Equation: Distinct Roots	117	(3)
7.3 The Auxiliary Equation: Repeated Roots	120	(3)
7.4 A Definition of exp Z for Imaginary Z	123	(2)
7.5 The Auxiliary Equation: Imaginary Roots	125	(2)
7.6 A Note on Hyperbolic Functions	127	(5)
7.7 Computer Supplement	132	(2)
8 Nonhomogeneous Equations: Undetermined Coefficients	134	(18)
8.1 Construction of a Homogeneous Equation from a Specific Solution	134	(3)
8.2 Solution of a Nonhomogeneous Equation	137	(2)
8.3 The Method of Undetermined Coefficients	139	(5)
8.4 Solution by Inspection	144	(6)
8.5 Computer Supplement	150	(2)
9 Variation of Parameters	152	(13)
9.1 Introduction	152	(1)
9.2 Reduction of Order	152	(4)
9.3 Variation of Parameters	156	(5)
9.4 Solution of y"+y=f(x)	161	(3)
9.5 Computer Supplement	164	(1)
Applications	165	(21)
10.1 Vibration of a Spring	165	(2)
10.2 Undamped Vibrations	167	(2)
10.3 Resonance	169	(3)
10.4 Damped Vibrations	172	(5)
10.5 The Simple Pendulum	177	(1)
10.6 Newton's Laws and Planetary Motion	178	(1)
10.7 Central Force and Kepler's Second Law	179	(1)
10.8 Kepler's First Law	180	(2)
10.9 Kepler's Third Law	182	(2)
10.10 Computer Supplement	184	(2)
11 Linear Systems of Equations	186	(38)
11.1 Introduction	186	(1)
11.2 First-Order Systems with Constant Coefficients	186	(1)
11.3 Solution of a First-Order System	187	(2)
11.4 Some Matrix Algebra	189	(6)
11.5 First-Order Systems Revisited	195	(9)
11.6 Complex Eigenvalues	204	(4)
11.7 Repeated Eigenvalues	208	(8)
11.8 The Phase Plane	216	(6)
11.9 Computer Supplement	222	(2)
12 Nonhomogeneous Systems of Equations	224	(19)
12.1 Nonhomogeneous Systems	224	(4)
12.2 Arms Races	228	(4)
12.3 Electric Circuits	232	(3)
12.4 Simple Networks	235	(8)
13 The Existence and Uniqueness of Solutions	243	(9)
13.1 Preliminary Remarks	243	(1)
13.2 An Existence and Uniqueness Theorem	243	(3)
13.3 A Lipschitz Condition	246	(1)
13.4 A Proof of the Existence Theorem	246	(4)
13.5 A Proof of the Uniqueness Theorem	250	(1)
13.6 Other Existence Theorems	251	(1)
14 The Laplace Transform	252	(22)
14.1 The Transform Concept	252	(1)
14.2 Definition of the Laplace Transform	253	(1)
14.3 Transforms of Elementary Function	253	(4)
14.4 Sectionally Continuous Functions	257	(1)
14.5 Functions of Exponential Order	258	(3)
14.6 Functions of Class A	261	(2)
14.7 Transforms of Derivatives	263	(3)
14.8 Derivatives of Transforms	266	(1)
14.9 The Gamma Function	267	(2)
14.10 Periodic Functions	269	(5)
15 Inverse Transforms	274	(46)
15.1 Definition of an Inverse Transform	274	(3)
15.2 Partial Fractions	277	(3)
15.3 Initial Value Problems	280	(6)
15.4 A Step Function	286	(8)
15.5 A Convolution Theorem	294	(4)
15.6 Special Integral Equations	298	(5)
15.7 Transform Methods and the Vibration of Springs	303	(4)
15.8 The Deflection of Beams	307	(3)
15.9 Systems of Equations	310	(6)
15.10 Computer Supplement	316	(4)
16 Nonlinear Equations	320	(22)
16.1 Preliminary Remarks	320	(1)
16.2 Factoring the Left Member	320	(3)
16.3 Singular Solutions	323	(2)
16.4 The c-Discriminant Equation	325	(1)
16.5 The p-Discriminant Equation	326	(2)
16.6 Eliminating the Dependent Variable	328	(2)
16.7 Clairaut's Equation	330	(4)
16.8 Dependent Variable Missing	334	(1)
16.9 Independent Variable Missing	335	(3)
16.10 The Catenary	338	(4)
17 Power Series Solutions	342	(16)
17.1 Linear Equations and Power Series	342	(1)
17.2 Convergence of Power Series	343	(2)
17.3 Ordinary Points and Singular Points	345	(2)
17.4 Validity of the Solutions Near an Ordianry Point	347	(1)
17.5 Solutions Near an Ordinary Point	347	(9)
17.6 Computer Supplement	356	(2)
18 Solutions Near Regular Singular Points	358	(38)
18.1 Regular Singular Points	358	(2)
18.2 The Indicial Equation	360	(2)
18.3 Form and Validity of the Solutions Near a Regular Singular Point	362	(1)
18.4 Indicial Equation with Difference of Roots Nonintegral	363	(4)
18.5 Differentiation of a Product of Functions	367	(1)
18.6 Indicial Equation with Equal Roots	368	(6)
18.7 Indicial Equation with Equal Roots: An Alternative	374	(3)
18.8 Indicial Equation with Difference of Roots a Positive Integer: Nonlogarithmic Case	377	(4)
18.9 Indicial Equation with Difference Roots a Positive Integer: Logarithmic Case	381	(4)
18.10 Solution for Large x	385	(3)
18.11 Many-Term Recurrence Relations	388	(4)
18.12 Summary	392	(4)
19 Equations of Hypergeometric Type	396	(8)
19.1 Equations to Be Treated in This Chapter	396	(1)
19.2 The Factorial Function	396	(1)
19.3 The Hypergeometric Function	397	(2)
19.4 Laguerre Polynomials	399	(1)
19.5 Bessel's Equation with Index Not an Integer	400	(1)
19.6 Bessel's Equation with Index an Integer	401	(1)
19.7 Hermite Polynomials	402	(1)
19.8 Legendre Polynomials	403	(1)
20 Partial Differential Equations	404	(14)
20.1 Remarks on Partial Differential Equations	404	(1)
20.2 Some Partial Differential Equations of Applied Mathematics	404	(2)
20.3 Method of Separation of Variables	406	(5)
20.4 A Problem on the Conduction of Heat in a Slab	411	(5)
20.5 Computer Supplement	416	(2)
21 Orthogonal Sets of Functions	418	(7)
21.1 Orthogonality	418	(1)
21.2 Simple Sets of Polynomials	419	(1)
21.3 Orthogonal Polynomials	419	(2)
21.4 Zeros of Orthogonal Polynomials	421	(1)
21.5 Orthogonality of Legendre Polynomials	422	(2)
21.6 Other Orthogonal Sets	424	(1)
22 Fourier Series	425	(22)
22.1 Orthogonality of a Set of Sines and Cosines	425	(2)
22.2 Fourier Series: An Expansion Theorem	427	(4)
22.3 Numerical Examples of Fourier Series	431	(7)
22.4 Fourier Sine Series	438	(2)
22.5 Fourier Cosine Series	441	(2)
22.6 Numerical Fourier Analysis	443	(1)
22.7 Improvement in Rapidity of Convergence	444	(1)
22.8 Computer Supplement	445	(2)
23 Boundary Value Problems	447	(20)
23.1 The One-Dimensional Heat Equation	447	(6)
23.2 Experimental Verification of the Validity of the Heat Equation	453	(2)
23.3 Surface Temperature Varying with Time	455	(2)
23.4 Heat Conduction in a Sphere	457	(1)
23.5 The Simple Wave Equation	458	(3)
23.6 Laplace's Equation in Two Dimensions	461	(3)
23.7 Computer Supplement	464	(3)
24 Additional Properties of the Laplace Transform	467	(14)
24.1 Power Series and Inverse Transforms	467	(4)
24.2 The Error Function	471	(7)
24.3 Bessel Functions	478	(2)
24.4 Differential Equations with Variable Coefficients	480	(1)
25 Partial Differential Equations Transform Methods	481	(19)
25.1 Boundary Value Problems	481	(4)
25.2 The Wave Equation	485	(3)
25.3 Diffusion in a Semi-Infinite Solid	488	(3)
25.4 Canonical Variables	491	(2)
25.5 Diffusion in a Slab of Finite Width	493	(3)
25.6 Diffusion in a Quarter-Infinite Solid	496	(4)
Answers to Odd-numbered Exercises	500	(27)
Index	527