Fundamentals of Differential Equations

by Nagle, R. Kent; Saff, E. B.; Snider, Arthur David

• ISBN13: 9780201338683
• ISBN10: 0201338688
• Edition: 5th
• Format: Hardcover w/CD
• Copyright: 2000-01-01
• Publisher: Addison Wesley

Summary

This text is in a flexible one-semester text that spans a variety of topics in the basic theory as well as applications of differential equations.

Table of Contents

Introduction	1	(41)
Background	1	(5)
Solutions and Initial Value Problems	6	(10)
Direction Fields	16	(8)
The Phase Line	24	(7)
The Approximation Method of Euler	31	(11)
Chapter Summary	37	(1)
Technical Writing Exercises	37	(1)
Group Projects for Chapter 1	38	(1)
Taylor Series Method	38	(1)
Picard's Method	39	(1)
Magnetic Dipole	40	(2)
First Order Differential Equations	42	(51)
Introduction: Motion of a Falling Body	42	(3)
Separable Equations	45	(9)
Linear Equations	54	(9)
Exact Equations	63	(10)
Special Integrating Factors	73	(4)
Substitutions and Transformations	77	(16)
Chapter Summary	86	(1)
Review Problems	87	(1)
Technical Writing Exercises	88	(1)
Group Projects for Chapter 2	89	(1)
The Snowplow Problem	89	(1)
Two Snowplows	89	(1)
Asymptotic Behavior of Solutions to Linear Equations	90	(1)
Torricelli's Law of Fluid Flow	90	(2)
Clairaut Equations and Singular Solutions	92	(1)
Mathematical Models and Numerical Methods Involving First Order Equations	93	(63)
Mathematical Modeling	93	(2)
Compartmental Analysis	95	(12)
Heating and Cooling of Buildings	107	(7)
Newtonian Mechanics	114	(10)
Improved Euler's Method	124	(11)
Higher-Order Numerical Methods: Taylor and Runge-Kutta	135	(9)
Professional Codes for Solving Initial Value Problems	144	(12)
Group Projects for Chapter 3	148	(1)
Delay Differential Equations	148	(1)
Aquaculture	149	(1)
Curve of Pursuit	150	(1)
Aircraft Guidance in a Crosswind	151	(1)
Stability of Numerical Methods	152	(1)
Period Doubling and Chaos	153	(2)
Bang-Bang Controls	155	(1)
Linear Second Order Equations	156	(105)
Introduction: The Mass-Spring Oscillator	156	(5)
Linear Differential Operators	161	(7)
Fundamental Solutions of Homogeneous Equations	168	(10)
Reduction of Order	178	(5)
Homogeneous Linear Equations with Constant Coefficients	183	(8)
Auxiliary Equations with Complex Roots	191	(9)
Superposition and Nonhomogeneous Equations	200	(4)
Method of Undetermined Coefficients	204	(9)
Variation of Parameters	213	(5)
Qualitative Considerations for Variable-Coefficient and Nonlinear Equations	218	(12)
A Closer Look at Free Mechanical Vibrations	230	(10)
A Closer Look at Forced Mechanical Vibrations	240	(21)
Chapter Summary	248	(2)
Review Problems	250	(1)
Technical Writing Exercises	251	(1)
Group Projects for Chapter 4	252	(1)
Undetermined Coefficients Using Complex Arithmetic	252	(1)
An Alternative to the Method of Undetermined Coefficients	253	(1)
Convolution Method	254	(1)
Linearization of Nonlinear Problems	255	(1)
Nonlinear Equations Solvable by First Order Techniques	256	(1)
Apollo Reentry	257	(1)
Simple Pendulum	258	(1)
Asymptotic Behavior of Solutions	259	(2)
Introduction to Systems and Phase Plane Analysis	261	(77)
Interconnected Fluid Tanks	261	(2)
Introduction to the Phase Plane	263	(15)
Elimination Method for Systems	278	(8)
Coupled Mass-Spring Systems	286	(7)
Electrical Circuits	293	(8)
Numerical Methods for Higher-Order Equations and Systems	301	(14)
Dynamical Systems, Poincare Maps, and Chaos	315	(23)
Chapter Summary	325	(2)
Review Problems	327	(1)
Group Projects for Chapter 5	328	(1)
Designing a Landing System for Interplanetary Travel	328	(1)
Things That Bob	329	(1)
Effects of Hunting on Predator-Prey Systems	330	(1)
Periodic Solutions to Volterra-Lotka Systems	331	(1)
Hamiltonian Systems	332	(2)
Limit Cycles	334	(1)
Strange Behavior of Competing Species---Part I	335	(1)
Cleaning Up the Great Lakes	336	(2)
Theory of Higher-Order Linear Differential Equations	338	(31)
Basic Theory of Linear Differential Equations	338	(9)
Homogeneous Linear Equations with Constant Coefficients	347	(7)
Undetermined Coefficients and the Annihilator Method	354	(6)
Method of Variation of Parameters	360	(9)
Chapter Summary	364	(2)
Review Problems	366	(1)
Technical Writing Exercises	366	(1)
Group Projects for Chapter 6	367	(1)
Justifying the Method of Undetermined Coefficients	367	(1)
Transverse Vibrations of Beam	367	(2)
Laplace Transforms	369	(78)
Introduction: A Mixing Problem	369	(4)
Definition of the Laplace Transform	373	(9)
Properties of the Laplace Transform	382	(6)
Inverse Laplace Transform	388	(10)
Solving Initial Value Problems	398	(8)
Transforms of Discontinuous and Periodic Functions	406	(14)
Convolution	420	(9)
Impulses and the Dirac Delta Function	429	(7)
Solving Linear Systems with Laplace Transforms	436	(11)
Chapter Summary	439	(1)
Review Problems	440	(1)
Technical Writing Exercises	441	(2)
Group Projects for Chapter 7	443	(1)
Duhamel's Formulas	443	(1)
Frequency Response Modeling	444	(2)
Determining System Parameters	446	(1)
Series Solutions of Differential Equations	447	(80)
Introduction: The Taylor Polynomial Approximation	447	(6)
Power Series and Analytic Functions	453	(9)
Power Series Solutions to Linear Differential Equations	462	(11)
Equations with Analytic Coefficients	473	(6)
Cauchy-Euler (Equidimensional) Equations Revisited	479	(4)
Method of Frobenius	483	(12)
Finding a Second Linearly Independent Solution	495	(12)
Special Functions	507	(20)
Chapter Summary	520	(1)
Review Problems	521	(1)
Technical Writing Exercises	522	(1)
Group Projects for Chapter 8	523	(1)
Spherically Symmetric Solutions to Schrodinger's Equation for the Hydrogen Atom	523	(1)
Airy's Equation	524	(1)
Buckling of a Tower	524	(1)
Aging Spring and Bessel Functions	525	(2)
Matrix Methods for Linear Systems	527	(73)
Introduction	527	(5)
Linear Algebraic Equations	532	(4)
Matrices and Vectors	536	(12)
Linear Systems in Normal Form	548	(9)
Homogeneous Linear Systems with Constant Coefficients	557	(12)
Complex Eigenvalues	569	(6)
Nonhomogeneous Linear Systems	575	(7)
The Matrix Exponential Function	582	(18)
Chapter Summary	591	(3)
Review Problems	594	(1)
Technical Writing Exercises	595	(1)
Group Projects for Chapter 9	596	(1)
Uncoupling Normal Systems	596	(1)
Matrix Laplace Transform Method	596	(2)
Undamped Second Order Systems	598	(1)
Strange Behavior of Competing Species---Part II	599	(1)
Partial Differential Equations	600
Introduction: A Model for Heat Flow	600	(3)
Method of Separation of Variables	603	(10)
Fourier Series	613	(18)
Fourier Cosine and Sine Series	631	(5)
The Heat Equation	636	(13)
The Wave Equation	649	(13)
Laplace's Equation	662
Chapter Summary	675	(2)
Technical Writing Exercises	677	(1)
Group Projects for Chapter 10	678	(1)
Steady-State Temperature Distribution in a Circular Cylinder	678	(1)
A Laplace Transform Solution of the Wave Equation	679	(1)
Green's Function	680	(2)
Numerical Method for Δu = f on a Rectangle	682
APPENDICES	A-1	(1)
A. Newton's Method	A-1	(1)
B. Simpson's Rule	A-3	(1)
C. Cramer's Rule	A-5	(1)
D. Method of Least Squares	A-6	(1)
E. Runge-Kutta Procedure for n Equations	A-9	(1)
Answers to Odd-Numbered Problems	B-1	(1)
Index	I-1