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
Contents
Preface
1
/
1
Examples of Differential Equations / 1
Definitions / 2
/ '
Families of Solutions / 5
Geometric Interpretation / 10
The Isoclines of an Equation / 12
An Existence Theorem / 14
Computer Supplement / 15
Equations of Order One
2.1
2.2
2.3
2.4
2.5
2.6
2.7
3
xiii
Definitions,- Families of Curves
1.1
1.2
1.3
1.4
1.5
1.6
1.7
2
/
/
18
Separation of Variables / 18
Homogeneous Functions / 24
Equations with Homogeneous Coefficients / 25
Exact Equations / 29
The Linear Equation of Order One / 35
The General Solution of a Linear Equation / 38
Computer Supplement / 43
Numerical Methods
45
3.1
General Remarks / 45
3.2
Euler's Method / 45
3.3 . A Modification of Euler's Method
/ 48
Contents
3.4
3.5
3.6
3.7
3.8
3.9
A Method of Successive Approximation
An Improvement on the Method
- o f Successive Approximation / 51
The Use of Taylor's Theorem / 52
The Runge-Kutta Method / ^ 54
A Continuing Method / 58
Computer Supplement / 60
Elementary Applications
4.1
4.2
4.3
4.4
4.5
/
/
49
62
Velocity of Escape from the Earth / 62
Newton's Law of Cooling / 64
Simple Chemical Conversion / 65
Logistic Growth and the Price of Commodities
Computer Supplement / 73
Additional Topics on Equations of Order One
5.1
5.2
5.3
5.4
5.5
5.6
5.7
69
/ 75
Integrating Factors Found by Inspection / 75
The Determination of Integrating Factors / 79
Substitution Suggested by the Equation / 83
Bernoulli's Equation / 86
Coefficients Linear in the Two Variables / 89
Solutions Involving Nonelementary Integrals / 94
Computer Supplement / 97
Linear Differential Equations
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
/
/
99
The General Linear Equation / 99
An Existence and Uniqueness Theorem / 100
Linear Independence / 102
The Wronskian / 103
General Solution of a Homogeneous Equation / 106
General Solution of a Nonhomogeneous Equation / 107
Differential Operators / 109
The Fundamental Laws of Operation / 111
Some Properties of Differential Operators / 113
Computer Supplement / 115
Contents
7
Linear Equations with Constant Coefficients
7.1
""7.27.3
7.4
7.5
7.6
7.7
8
8.2
8.3
8.4
8.5
9
/
152
Introduction / 152
Reduction of Order / 152
Variation of Parameters / 156
Solution of y"+y=f(x)
I 161
Computer Supplement / 164
Applications
10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8
134
Construction of a Homogeneous Equation
from a Specific Solution / 134 '
Solution of a Nonhomogeneous Equation / 137
The Method of Undetermined Coefficients / 139
Solution by Inspection / 144
Computer Supplement / 150
Variation of Parameters
9.1
9.2
9.3
9.4
9.5
10
Introduction / 117
The Auxiliary Equation: Distinct Roots / 117
The Auxiliary Equation: Repeated Roots / 120
A Definition of exp z for Imaginary z I 123
The Auxiliary Equation: Imaginary Roots / 125
A Note on Hyperbolic Functions / 127
Computer Supplement / 132
Nonhomogeneous Equations:
Undetermined Coefficients /
8.1
/
/
165
Vibration of a Spring / 165
Undamped Vibrations / 167
Resonance / 169
Damped Vibrations / 172
The Simple Pendulum / 177
Newton's Laws and Planetary Motion / 178
Central Force and Kepler's Second Law / 179
Kepler's First Law / 180
117
Vlll
Contents
10.9
Kepler's Third Law / 182
10.10 Computer Supplement / 184
11
~~"
\
Linear Systems of Equations /
11.1
11.2
11.3
11.4
11.5
11.6
11.7
11.8
11.9
12
Nonhomogeneous Systems /
Arms Races / 228
Electric Circuits / 232
Simple Networks / 235
/
/
186
224
/
Preliminary Remarks / 243
An Existence and Uniqueness Theorem / 243
A Lipschitz Condition / 246
A Proof of the Existence Theorem / 246
A Proof of the Uniqueness Theorem / 250
Other Existence Theorems / 251
The Laplace Transform
14.1
14.2
14.3
14.4
14.5^
14.6
/
224
The Existence and Uniqueness of Solutions
13.1
___ -13.2
13.3
13.4
13.5
13.6
14
Introduction / 186
First-Order Systems with Constant Coefficients
Solution of a First-Order System / 187
Some Matrix Algebra / 189
First-Order Systems Revisited / 195
Complex Eigenvalues / 204
Repeated Eigenvalues / 208
The Phase Plane / 216
Computer Supplement / 222
Nonhomogeneous Systems of Equations
12.1
12.2
12.3
12.4
13
186
252
The Transform Concept / 252
Definition of the Laplace Transform / 253
Transforms of Elementary Functions / 253
Sectionally Continuous Functions / 257
Functions of Exponential Order / 258
Functions of Class A / 261
243
Contents
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
15.1
15.2
15.3
15.4
15.5
15.6
15.7
15.8
15.9
15.10
16
/
274
Definition of an Inverse Transform / 274
Partial Fractions / 277
Initial Value Problems / 280
A Step Function / 286
A Convolution Theorem
/ 294
Special Integral Equations / 298
Transform Methods and the Vibration offsprings
The Deflection of Beams / 307 ""
Systems of Equations / 310
Computer Supplement / 316
Nonlinear Equations
/
/
303
/
347
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
47
Power Series Solutions
17.1
17.2
17.3
17.4
17.5
17.6
/
342
Linear Equations and Power Series / 342
Convergence of Power Series / 343
Ordinary Points and Singular Points / 345
Validity of the Solutions Near an Ordinary Point
Solutions Near an Ordinary Point / 347
Computer Supplement / 356
Contents
\
18
Solutions Near Regular Singular Points
/
358
18.1
18:2
18.3
Regular Singular Points / 358
The Indicial Equation / 360
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
~"
20
19.1
19.2
19.3
19.4
19.5
19.6
19.7
19.8
20.3
396
Equations to Be Treated in This Chapter / 396
The Factorial Function / 396
The Hypergeometric Function / 397
Laguerre Polynomials / 399
Bessel's Equation with Index Not an Integer / 400
Bessel's Equation with Index an Integer / 401
Hermite Polynomials / 402
Legendre Polynomials / 403
Partial Differential Equations
20.1
20.2
/
/
404
Remarks on Partial Differential Equations /
Some Partial Differential Equations
of Applied Mathematics / 404
Method of Separation of Variables / 406
404
\
Contents
xi
20.4
20.5
21
Orthogonal Sets of Functions
21.1
21.2
21.3
21.4
21.5
21.6
22
23.3
23.4
23.5
23.6
23.7
24
/
411
418
422
Orthogonality of a Set of Sines and Cosines-'/ 425
Fourier Series: An Expansion Theorem / 427
Numerical Examples of Fourier Series / 431
Fourier Sine Series / 438
Fourier Cosine Series / 441
Numerical Fourier Analysis / 443
Improvement in Rapidity of Convergence / 444
Computer Supplement / 445
/
447
The One-Dimensional Heat Equation / 447
Experimental Verification of the Validity
of the Heat Equation / 453
Surface Temperature Varying with Time / 455
Heat Conduction in a Sphere / 457
The Simple Wave Equation / 458
Laplace's Equation in Two Dimensions / 461
Computer Supplement / 464
Additional Properties of the Laplace Transform
24.1
24.2
/
425
Boundary Value Problems
23.1
23.2
/
Orthogonality / 418
Simple Sets of Polynomials / 419
Orthogonal Polynomials / 419
Zeros of Orthogonal Polynomials / 421
Orthogonality of Legendre Polynomials /
Other Orthogonal Sets / 424
Fourier Series
22.1
22.2
22.3
22.4
22.5
22.6
22.7
22.8
23
A Problem on the Conduction of Heat in a Slab
Computer Supplement / 416
Power Series and Inverse Transforms
The Error Function / 471
/
467
/
467
Contents
24.3
24.4
25
Bessel Functions / 478
Differential Equations with Variable Coefficients
/
Partial Differential Equations Transform Methods
25.1
25.2
25.3
25.4
25.5
25.6
Boundary Value Problems / 481
The Wave Equation / 485
Diffusion in a Semi-Infinite Solid / 488
Canonical Variables / 491
Diffusion in a Slab of Finite Width / 493
Diffusion in a Quarter-Infinite Solid / 496
Answers to Odd-numbered Exercises
Index
/
527
/
500
480
/
481