Elementary Differential Equations

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1 Elementary Differential Equations EIGHTH EDITION Earl D. Rainville Late Professor of Mathematics University of Michigan Phillip E. Bedient Professor Emeritus of Mathematics Franklin and Marshall College Richard E. Bedient Professor of Mathematics Hamilton College PRENTICE HALL, UPPER SADDLE RIVER, NJ 07458

2 Preface / xiii 1 Definitions,- Families of Curves / Examples of Differential Equations / Definitions / 2 / ' 1.3 Families of Solutions / Geometric Interpretation / The Isoclines of an Equation / An Existence Theorem / Computer Supplement / 15 2 Equations of Order One / Separation of Variables / Homogeneous Functions / Equations with Homogeneous Coefficients / Exact Equations / The Linear Equation of Order One / The General Solution of a Linear Equation / Computer Supplement / 43 3 Numerical Methods General Remarks / Euler's Method / A Modification of Euler's Method / 48

3 3.4 A Method of Successive Approximation / An Improvement on the Method -of Successive Approximation / The Use of Taylor's Theorem / The Runge-Kutta Method /^ A Continuing Method / Computer Supplement / 60 Elementary Applications / Velocity of Escape from the Earth / Newton's Law of Cooling / Simple Chemical Conversion / Logistic Growth and the Price of Commodities / Computer Supplement / 73 Additional Topics on Equations of Order One / Integrating Factors Found by Inspection / The Determination of Integrating Factors / Substitution Suggested by the Equation / Bernoulli's Equation / Coefficients Linear in the Two Variables / Solutions Involving Nonelementary Integrals / Computer Supplement / 97 Linear Differential Equations / The General Linear Equation / An Existence and Uniqueness Theorem / Linear Independence / The Wronskian / General Solution of a Homogeneous Equation / General Solution of a Nonhomogeneous Equation / Differential Operators / The Fundamental Laws of Operation / Some Properties of Differential Operators / Computer Supplement / 115

4 7 Linear Equations with Constant Coefficients / Introduction / 117 ""7.2- The Auxiliary Equation: Distinct Roots / The Auxiliary Equation: Repeated Roots / A Definition of exp z for Imaginary z I The Auxiliary Equation: Imaginary Roots / A Note on Hyperbolic Functions / Computer Supplement / Nonhomogeneous Equations: Undetermined Coefficients / Construction of a Homogeneous Equation from a Specific Solution / 134 ' 8.2 Solution of a Nonhomogeneous Equation / The Method of Undetermined Coefficients / Solution by Inspection / Computer Supplement / Variation of Parameters / Introduction / Reduction of Order / Variation of Parameters / Solution of y"+y=f(x) I Computer Supplement / Applications / Vibration of a Spring / Undamped Vibrations / Resonance / Damped Vibrations / The Simple Pendulum / Newton's Laws and Planetary Motion / Central Force and Kepler's Second Law / Kepler's First Law / 180

5 Vlll Contents 10.9 Kepler's Third Law / Computer Supplement / 184 ~~" \ 11 Linear Systems of Equations / Introduction / First-Order Systems with Constant Coefficients / Solution of a First-Order System / Some Matrix Algebra / First-Order Systems Revisited / Complex Eigenvalues / Repeated Eigenvalues / The Phase Plane / Computer Supplement / Nonhomogeneous Systems of Equations / Nonhomogeneous Systems / Arms Races / Electric Circuits / Simple Networks / The Existence and Uniqueness of Solutions / Preliminary Remarks / An Existence and Uniqueness Theorem / A Lipschitz Condition / A Proof of the Existence Theorem / A Proof of the Uniqueness Theorem / Other Existence Theorems / The Laplace Transform / The Transform Concept / Definition of the Laplace Transform / Transforms of Elementary Functions / Sectionally Continuous Functions / ^ Functions of Exponential Order / Functions of Class A / 261

6 IX 14.7 Transforms of Derivatives / Derivatives of Transforms / The Gamma Function / Periodic Functions / Inverse Transforms / Definition of an Inverse Transform / Partial Fractions / Initial Value Problems / A Step Function / A Convolution Theorem / Special Integral Equations / Transform Methods and the Vibration offsprings / The Deflection of Beams / 307 "" 15.9 Systems of Equations / Computer Supplement / Nonlinear Equations / Preliminary Remarks / Factoring the Left Member / Singular Solutions / The c-discriminant Equation / The p-discriminant Equation / Eliminating the Dependent Variable / Clairaut's Equation / Dependent Variable Missing / Independent Variable Missing / The Catenary / Power Series Solutions / Linear Equations and Power Series / Convergence of Power Series / Ordinary Points and Singular Points / Validity of the Solutions Near an Ordinary Point / Solutions Near an Ordinary Point / Computer Supplement / 356

7 \ 18 Solutions Near Regular Singular Points / Regular Singular Points / :2 The Indicial Equation / Form and Validity of the Solutions Near a Regular Singular Point / Indicial Equation with Difference of Roots Nonintegral / Differentiation of a Product of Functions / Indicial Equation with Equal Roots / Indicial Equation with Equal Roots: An Alternative / Indicial Equation with Difference of Roots a Positive Integer: Nonlogarithmic Case^- / Indicial Equation with Difference of Roots a Positive Integer: Logarithmic Case / Solution for Large x I Many-Term Recurrence Relations / Summary / Equations of Hypergeometric Type / Equations to Be Treated in This Chapter / The Factorial Function / The Hypergeometric Function / Laguerre Polynomials / 399 ~" 19.5 Bessel's Equation with Index Not an Integer / Bessel's Equation with Index an Integer / Hermite Polynomials / Legendre Polynomials / Partial Differential Equations / Remarks on Partial Differential Equations / Some Partial Differential Equations of Applied Mathematics / Method of Separation of Variables / 406

8 \ 20.4 A Problem on the Conduction of Heat in a Slab / Computer Supplement / 416 xi 21 Orthogonal Sets of Functions / Orthogonality / Simple Sets of Polynomials / Orthogonal Polynomials / Zeros of Orthogonal Polynomials / Orthogonality of Legendre Polynomials / Other Orthogonal Sets / Fourier Series / Orthogonality of a Set of Sines and Cosines-'/ Fourier Series: An Expansion Theorem / Numerical Examples of Fourier Series / Fourier Sine Series / Fourier Cosine Series / Numerical Fourier Analysis / Improvement in Rapidity of Convergence / Computer Supplement / Boundary Value Problems / The One-Dimensional Heat Equation / Experimental Verification of the Validity of the Heat Equation / Surface Temperature Varying with Time / Heat Conduction in a Sphere / The Simple Wave Equation / Laplace's Equation in Two Dimensions / Computer Supplement / Additional Properties of the Laplace Transform / Power Series and Inverse Transforms / The Error Function / 471

9 24.3 Bessel Functions / Differential Equations with Variable Coefficients / 480 Contents 25 Partial Differential Equations Transform Methods / Boundary Value Problems / The Wave Equation / Diffusion in a Semi-Infinite Solid / Canonical Variables / Diffusion in a Slab of Finite Width / Diffusion in a Quarter-Infinite Solid / 496 Answers to Odd-numbered Exercises / 500 Index / 527

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