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