Ordinary Differential Equations and Special Functions
Description
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.
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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.
Content
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 (a,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
1
Preliminaries I
Abstract
This chapter discusses first-order and first-degree differential equations, as well as higher-order and higher-degree differential equations. We explore several special solution methods applicable to different types of equations and families of curves represented by differential equations. These methods are used throughout the first part of the book as ready references.
Keywords: Particular integral, Bernoulli's equation, clauriant's equation, total differential equation
1.1 Introduction
We begin our study of differential equations by explaining what a differential equation is. From our experience in calculus, we are familiar with some differential equations. For example, suppose that the acceleration due to gravity, a(t) (measured in ft/s2 ), of a falling object is a(t) = -14. Then, because a(t) = v´(t), where v(t) is the velocity of the object (measured in ft/s), we have v´(t) = 14 or . An equation like this involving a function of a single variable is called an ordinary differential equation (ODE). [If the equation involves partial derivatives, then it is called a partial differential equation (PDE).] In this case, the function to be determined is v = v(t), which depends on the variable t representing time (measured in seconds). The goal in solving an ODE is to find a function that satisfies the equation. We can solve this ODE through integration:
where c is an arbitrary constant.
This result indicates that v(t) = -14t + c is a solution of the ODE for any choice of the constant c. (We call this a general solution because it involves an arbitrary constant.) In fact, we have found every solution of the ODE because each is expressed as -14t + c. Examples of solutions and the corresponding c values include v(t) = -14t (c = 0), v(t) = -14t + 15(c = 15), and v(t) = -14t - 28 (c = -28). This shows that there are an infinite number of solutions to the ODE.
The following are all examples of differential equations:
An ordinary differential equation is that in which all the differential coefficients have one single independent variable. Thus, Equations (i-iii) are all ordinary differential equations.
A partial differential equation is that in which there are two or more independent variables and partial derivatives with respect to any of them. Equations (iv) and (v) are partial differential equations.
The degree and order of differential equations are the following:
(i) First degree and first order, (ii) second order and first degree, (iii) second order and second degree, (iv) first order, first degree, and (v) second order, first degree.
This chapter also discusses on method of solving different differential equations and studies the method of variation of parameters, orthogonal trajectories and total differential equations. It also discusses the results for finding the particular integral which are used frequently later on.
1.2 Formation of a Differential Equation
For an applied mathematician, the study of a differential equation consists of three phases as follows: i) formation of differential equation from a given physical situation, which is called Modeling, ii) solutions of this equation by evaluating the arbitrary conditions, and iii) physical interpretation of the solutions.
Example 1.2.1. The distance s covered in time t by a free particle of fixed mass m under the influence of a force P, is by Newton's second law of motion such that
s is an unknown function of independent variable t, and P is a function of s (physically it is force) and/or t. In a particular case, we consider the motion of a particle projected vertically upward from the surface under gravitational attraction so that
The acceleration of the particle is given by
By chain rule of differentiation, we get
Here, x = R and acceleration .
So that GM = gR2. The equation will be , which is a differential equation in an unknown v, a function of s.
Example 1.2.2. Formation of the differential equation of a simple harmonic equation:
Solution: Eliminating k and ?, differentiating twice, we get
Hence, .
Definition 1.2.1. If a differential equation is of first degree in the dependent variable x and its derivative, consequently, there cannot be any term involving the product of x and its derivatives, then it is called a linear differential equation; otherwise, it is non-linear.
Example 1.2.3. Obtain the differential equation corresponding to the primitive
(i)Solution: Differentiating the equation with respect to x, we get
(ii)Differentiating
or,
(iii)From Equation (iii), .
Putting these values in the equation
(iv)Note. If a, ß varies in (ia) we get a system of circles of a given radius a having their centers anywhere in the plane. Equation (iii) expresses that, for every member of the system, the radius of curvature has everywhere the constant value a.
Remark 1.2.1. Independence of the n arbitrary constant a in a general solution, means that the solution cannot be reduced to a form containing (n-1) or fewer constants.
Let
be a solution of the equation , containing two arbitrary constant c1 and c2.
But, it is not a general solution, since it can be written as
which contains only one constant c.
Example 1.2.4. Consider the differential equation
(i)Suppose, by the same means, we obtain from it the relation
(ii)It is a solution of Equation (i). For putting y = e2x, we get . We found that Equation (1.1) is identically satisfied. But it may not be the only solution if we verify that y = e-2x is also solution. Generally, the relation y = Ae2x + Be2x also satisfies Equation (i): A, B are independent, arbitrary constants.
The solution of a differential equation, which contains the same number of independent arbitrary constants as the order of the differential equation, is called its general solution (primitive or complete integral or complete).
Any solution obtained from the general solution by giving particular values to the arbitrary constants is called a particular solution.
The general solution does not always include all possible solutions of a differential equation. Solutions may exist, which are not deducible from the general solution by giving particular values to the constants. Such solutions are called singular solutions.
The equation
(iii)where a and b are constants, is satisfied by
(iv)where c is an arbitrary constant. This is a general solution of Equation (iii).
But the relation
(v)also satisfies Equation (iii) but not deducible from Equation (iv). Hence, Equation (v) is a singular solution of Equation (iii).
Since the general solution of an nth-order equation contains n independent arbitrary constants, the maximum number of independent conditions, which can be imposed on a solution, is n. If these conditions pertain to the behavior of the solution at one specific point, they are called initial conditions, and the differential equation with such conditions is called an initial value problem (IVP). On the other hand, a differential equation with a condition prescribed at more than one point is called a boundary value problem (BVP).
For example: , y(0)=0 and is an example of IVP and y = sin2x is the solution of the problem.
For BVP, we consider , y(0)=1, .
It can be seen that y = cosx + 2sinx is a solution of the above BVP.
1.3 Family of Curves Represented by Ordinary Differential Equations
The primitive of an ordinary differential equation of first order is a relation between the two variables x and y and a parameter c. For each numerical value of c, we get a separate curve, and hence, the primitive gives a one-parameter family of curves. The differential equation is then said to represent a one-parameter family of curves. Each curve of family is called an integral curve.
The equation of first order and first degree is
As we have seen earlier, a general solution of the nth order differential equation is
(1.1)If it exists, it can be written in the form
(1.2)c1, c2, . cn are arbitrary constants,
which represents an n-parameter family of curves.
For the nth parameter of the above family of curves...
