Ordinary Differential Equations and Special Functions

This book is an essential guide for anyone in engineering or mathematical physics looking to master the fundamental concepts of differential equations and special functions, which are crucial for solving real-world problems. In today’s evolving mathematics landscape, differential equations and special functions have shown great promise for applications in engineering. Problems in mathematical physics help determine solutions for differential equations under certain parameters, which can be turned into new special functions, such as Bessel’s functions, to measure electricity, hydrodynamics, and vibration. Ordinary Differential Equations and Special Functions serves as a fundamental guide to these concepts, covering everything from elementary-level special functions to differential equations with a series of solutions.

Table of Contents:
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 (α,β ,γ ;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

About the Author :
Dipankar De, PhD is an contractual professor and guest lecturer with more than 40 years of experience. He has published several research papers in various reputed journals in the fields of fuzzy mathematics and differential geometry.