Elementary Differential Equations - GBV

1 ElementaryDifferentialEquationsEIGHTH EDITIONEarl D. RainvilleLate Professor of MathematicsUniversity of MichiganPhillip E. BedientProfessor Emeritus of MathematicsFranklin and Marshall CollegeRichard E. BedientProfessor of MathematicsHamilton CollegePRENTICE HALL, UPPER SADDLE RIVER, NJ 07458 ContentsPreface / xiii1 Definitions,- Families of Curves / Examples of Differential Equations / Definitions / 2 /' Families of Solutions / Geometric Interpretation / The Isoclines of an Equation / An Existence Theorem / Computer Supplement / 152 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 / 433 Numerical Methods General Remarks / Euler's Method /.

2 A Modification of Euler's Method / 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 Contents7 Linear Equations with Constant Coefficients / Introduction / 117"" The Auxiliary Equation.

3 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 / 1328 Nonhomogeneous Equations :Undetermined Coefficients / Construction of a Homogeneous Equationfrom a Specific Solution / 134 ' Solution of a Nonhomogeneous Equation / The Method of Undetermined Coefficients / Solution by Inspection / Computer Supplement / 1509 Variation of Parameters / Introduction / Reduction of Order / Variation of Parameters / Solution of y"+y=f(x) I Computer Supplement / 16410 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 Vlll Kepler's Third Law / Computer Supplement / 184~~" \11 Linear Systems of Equations / Introduction / First-Order Systems with Constant Coefficients /

4 Solution of a First-Order System / Some Matrix Algebra / First-Order Systems Revisited / Complex Eigenvalues / Repeated Eigenvalues / The Phase Plane / Computer Supplement / 22212 Nonhomogeneous Systems of Equations / Nonhomogeneous Systems / Arms Races / Electric Circuits / Simple Networks / 23513 The Existence and Uniqueness of Solutions / Preliminary Remarks / 243___ An Existence and Uniqueness Theorem / A Lipschitz Condition / A Proof of the Existence Theorem / A Proof of the Uniqueness Theorem / Other Existence Theorems / 25114 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 Contents Transforms of Derivatives / Derivatives of Transforms / The Gamma Function / Periodic Functions / 26915 Inverse Transforms / Definition of an Inverse Transform / Partial Fractions / Initial Value Problems / A Step Function / A Convolution Theorem /

5 Special Integral Equations / Transform Methods and the Vibration offsprings / The Deflection of Beams / 307 "" Systems of Equations / Computer Supplement / 31616 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 / 3384 7 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 Contents\18 Solutions Near Regular Singular Points / Regular Singular Points / 35818:2 The Indicial Equation / Form and Validity of the SolutionsNear a Regular Singular Point / Indicial Equation with Differenceof 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 Rootsa Positive Integer: Nonlogarithmic Case^- / Indicial Equation with Difference of Rootsa Positive Integer.

6 Logarithmic Case / Solution for Large x I Many-Term Recurrence Relations / Summary / 39219 Equations of Hypergeometric Type / Equations to Be Treated in This Chapter / The Factorial Function / The Hypergeometric Function / Laguerre Polynomials / 399~" Bessel's Equation with Index Not an Integer / Bessel's Equation with Index an Integer / Hermite Polynomials / Legendre Polynomials / 40320 Partial Differential Equations / Remarks on Partial Differential Equations / Some Partial Differential Equationsof Applied Mathematics / Method of Separation of Variables / 406\Contents A Problem on the Conduction of Heat in a Slab / Computer Supplement / 41621 Orthogonal Sets of Functions / Orthogonality / Simple Sets of Polynomials / Orthogonal Polynomials / Zeros of Orthogonal Polynomials / Orthogonality of Legendre Polynomials / Other Orthogonal Sets / 42422 Fourier Series / Orthogonality of a Set of Sines and Cosines-'/ Fourier Series.

7 An Expansion Theorem / Numerical Examples of Fourier Series / Fourier Sine Series / Fourier Cosine Series / Numerical Fourier Analysis / Improvement in Rapidity of Convergence / Computer Supplement / 44523 Boundary Value Problems / The One-Dimensional Heat Equation / Experimental Verification of the Validityof 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 / 46424 Additional Properties of the Laplace Transform / Power Series and Inverse Transforms / The Error Function / Bessel Functions / Differential Equations with Variable Coefficients / 48025 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