Ordinary Differential Equations and Special Functions

Preface xiii
Introduction xv
About the Book xvii

Part I: Ordinary Differential Equations 1

1 Preliminaries I 3
1.1 Introduction 3
1.2 Formation of a Differential Equation 4
1.3 Family of Curves Represented by Ordinary Differential Equations 8
1.4 Equation of the First Order and First Degree 11
1.5 Equations of the First Order and Higher Degree 24
1.6 Linear Differential Equation 30
1.7 Other Methods of Finding P.I. 40
1.8 Differential Equation of Other Types 44
1.9 Orthogonal Trajectories 51
1.10 Examples 52
1.11 Exercise 63

2 Existence Theorems 67
2.1 Introduction 67
2.2 Initial Value Problems and Boundary Value Problems 69
2.3 Picard's Method of Successive Approximation 70
2.4 Lipschitz Condition 78
2.5 Picard's Theorem: Existence and Uniqueness Theorem 80
2.6 Singular Solutions 90
2.7 Clairaut Equation 92
2.8 Examples 95
2.9 Exercise 99

3 System of Linear Differential Equations-I 101
3.1 Introduction 101
3.2 Matrix Form of a Linear System 102
3.3 Reduction of an nth-Order Equation 104
3.4 Matrix Preliminaries 107
3.5 Fundamental Set of Solutions 109
3.6 Solution of Non-Homogeneous Linear Systems 126
3.7 Linear System with Constant Coefficients 130
3.8 Exercise 138

4 Systems of Linear Differential Equations-II 141
4.1 Introduction 141
4.2 Linearly Dependent and Independent Functions 144
4.3 The Second-Order Homogeneous Equation 151
4.4 Non-Homogeneous Equation of Second-Order: Method of Variation of Parameters 157
4.5 Higher-Order Homogeneous Linear Differential Equations with Constant Coefficients 163
4.6 Examples 174
4.7 Exercise 177

5 Adjoint Equation 181
5.1 Introduction 181
5.2 Adjoint Equation 181
5.3 Green's Formula 194
5.4 Examples 196
5.5 Exercise 201

6 Boundary Value Problem 203
6.1 Introduction 203
6.2 Green's Function 207
6.3 Examples 210
6.4 Exercise 215

7 Strum Liouville Problem 217
7.1 Introduction 217
7.2 Strum-Liouville Equation 217
7.3 Orthogonality of Eigen Functions 220
7.4 Orthonormal Set of Functions 222
7.5 Gram-Schmidt Process of Orthonormalization 222
7.6 Reality of Eigenvalues 225
7.7 Examples 229
7.8 Exercise 233

Part II: Special Functions 235

8 Preliminaries II 237
8.1 Introduction 237
8.2 Infinite Series 237
8.3 Infinite Integrals 242
8.4 Infinite Products 245
8.5 Some Theorems on Functions of Complex Variables 246
8.6 Exercise 250

9 Series Solution of Differential Equations 253
9.1 Introduction 253
9.2 Power Series 254
9.3 Power Series Solution Near the Ordinary Point x = x0 256
9.4 Series Solution About Regular Singular Point x = 0: Frobenius Method 261
9.5 Examples 269
9.6 Exercise 307

10 Hypergeometric Functions 309
10.1 Introduction 309
10.2 Differentiation of Hypergeometric Functions 314
10.3 An Integral Formula for a Hypergeometric Function 316
10.4 Transformation of F (¿,ß ,¿ ;x) 324
10.5 Hypergeometric Equation 328
10.6 Confluent Hypergeometric Series 335
10.7 Contiguous Hypergeometric Functions 342
10.8 Generalized Hypergeometric Series 344
10.9 Integrals Involving Generalized Hypergeometric Functions 355
10.10 Some Special Generalized Hypergeometric Functions 357
10.11 Barnes Type Contour Integrals 365
10.12 Example 367
10.13 Exercise 370

11 Bessel Functions 373
11.1 Introduction 373
11.2 Bessel's Equation 374
11.3 Recurrence Formulae for Jn(x) 378
11.4 Expansion of J0 , J1 ,and J1/2 382
11.5 Generating Function for Jn(x) 399
11.6 Modified Bessel Functions 411
11.7 Equations Reducible to Bessel Equation 416
11.8 Orthogonality of Bessel Functions 418
11.9 Zeros of Bessel Functions 424
11.10 Ber and Bei Functions 427
11.11 Exercise 428

12 Legendre Polynomials 431
12.1 Introduction 431
12.2 Legendre's Equation 432
12.3 Another Form of Legendre's Polynomial Pn(x) 435
12.4 Generating Function for Legendre's Polynomials 438
12.5 Various Forms of Pn(x) 442
12.6 Recurrence Formulae for Pn(x) 446
12.7 Christoffel's Summation Formula 448
12.8 Orthogonality of Legendre Polynomials 451
12.9 Fourier-Legendre's Expansion of f (x) 453
12.10 Associated Legendre's Functions 468
12.11 Legendre's Functions of the Second Kind-Qn(x) 481
12.12 Examples 488
12.13 Exercise 494

13 Hermite Polynomials 497
13.1 Introduction 497
13.2 Hermite Equation and Its Solution 497
13.3 Generating Function for Hermite Polynomials 502
13.4 Recurrence Relations 506
13.5 Orthogonal Property 511
13.6 Expansion of Polynomials 512
13.7 More Generating Functions 515
13.8 Examples 517
13.9 Exercise 522

14 Laguerre Polynomials 525
14.1 Introduction 525
14.2 Laguerre's Equation and Its Solution 525
14.3 Generating Function of Laguerre Polynomials 528
14.4 Orthogonality Properties of Laguerre Polynomials 531
14.5 Recurrence Relations 533
14.6 Expansion of Laguerre Polynomials 537
14.7 Properties of Laguerre Polynomials 539
14.8 Generalized Laguerre Polynomial 541
14.9 Examples 551
14.10 Exercise 558

15 Jacobi Polynomials 561
15.1 Introduction 561
15.2 Jacobi Polynomial 562
15.3 Generating Functions 564
15.4 Rodrigues' Formula 567
15.5 Orthogonality of Jacobi Polynomial 568
15.6 Recurrence Relations 572
15.7 Expansions 580
15.8 Examples 582
15.9 Exercise 585

16 Chebyshev Polynomials 587
16.1 Introduction 587
16.2 Chebyshev Polynomials 587
16.3 Orthogonality Property 591
16.4 Recurrence Relations 593
16.5 Identities of Chebyshev Polynomials 594
16.6 Expansions 595
16.7 Generating Function 598
16.8 Rodrigues Formula of Chebyshev Polynomials 599
16.9 Exercise 601

Appendix A: Answer to Even-Numbered Exercises 603

References 611
Index 613