EXACT SOLUTIONS for ORDINARY DIFFERENTIAL EQUATIONS SECOND EDITION

Transcription

1 HANDBOOK OF EXACT SOLUTIONS for ORDINARY DIFFERENTIAL EQUATIONS SECOND EDITION Andrei D. Polyanin Valentin F. Zaitsev CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C.

2 Authors ~. xxi Foreword Notations and Some Remarks xxiii xxv Introduction Some Definitions, Formulas, Methods, and Transformations First-Order Differential Equations General Concepts. The Cauchy Problem. Uniqueness and Existence Theorems Equations solved for the derivative. General solution The Cauchy problem. The uniqueness and existence theorems Equations not solved for the derivative. The existence theorem Singular solutions Point transformations Equations Solved for the Derivative. Simplest Techniques of Integration Equations with separated or separable variables Equation of the form y' x = f(ax + by) Homogeneous equations and equations reducible to them Generalized homogeneous equations and equations reducible to them Linear equation Bernoulli equation Equation of the form xy' x =y + f(x)g(y/x) Darboux equation Exact Differential Equations. Integrating Factor Exact differential equations Integrating factor Riccati Equation General Riccati equation. Simplest integrable cases Polynomial solutions of the Riccati equation Use of particular solutions to construct the general solution Some transformations Reduction of the Riccati equation to a second-order linear equation Reduction of the Riccati equation to the canonical form Abel Equations of the'first Kind General form of Abel equations of the first kind. Simplest integrable cases Reduction to the canonical form. Reduction to an Abel equation of the second kind Abel Equations of the Second Kind General form of Abel equations of the second kind. Simplest integrable cases Reduction to the canonical form. Reduction to an Abel equation of the firstkind Use of particular solutions to construct self-transformations Use of particular solutions to construct the general solution Equations Not Solved for the Derivative The method of "integration by differentiation." Equations of the form y = f(y' x ) 14

3 vi CONTENTS Equations of the form x = f{y' x ) Clairaut's equation y = xy' x + f(y' x ) Lagrange's equation y = xf(y' x ) + g(y' x ) Contact Transformations General form of contact transformations A method for the construction of contact transformations Examples of contact transformations linear in the derivative Examples of contact transformations nonlinear in the derivative Approximate Analytic Methods for Solution of Equations The method of successive approximations (Picard method) The method of Taylor series expansion in the independent variable The method of regular expansion in the small parameter Numerical Integration of Differential Equations The method of Euler polygonal lines Single-step methods of the second-order approximation Runge-Kutta method of the fourth-order approximation Second-Order Linear Differential Equations Formulas for the General Solution. Some Transformations Homogeneous linear equations. Various representations of the general solution Wronskian determinant and Liouville's formula Reduction to the canonical form Reduction to the Riccati equation Nonhomogeneous linear equations. The existence theorem Nonhomogeneous linear equations. Various representations of the general solution Reduction to a constant coefficient equation (a special case) Kummer-Liouville transformation Representation of Solutions as a Series in the Independent Variable Equation coefficients are representable in the ordinary power series form Equation coefficients have poles at some point Asymptotic Solutions Equations not containing y' x. Leading asymptotic terms Equations not containing y' x. Two-term asymptotic expansions Equations of special form not containing y' x Equations not containing y' x. Equation coefficients are dependent on e Equations containing y' x Equations of the general form Boundary Value Problems The first, second, third, and mixed boundary value problems Simplification of boundary conditions. Reduction of equation to the self-adjoint form The Green's function. Boundary value problems for nonhomogeneous equations Representation of the Green's function in terms of particular solutions Eigenvalue Problems The Sturm-Liouville problem General properties of the Sturm-Liouville problem (1), (2) Problems with boundary conditions of the first kind Problems with boundary conditions of the second kind 32

4 vii Problems with boundary conditions of the third kind Problems with mixed boundary conditions Second-Order Nonlinear Differential Equations Form of the General Solution. Cauchy Problem Equations solved for the derivative. General solution Cauchy problem. The existence and uniqueness theorem Equations Admitting Reduction of Order Equations not containing y explicitly Equations not containing x explicitly (autonomous equations) Equations of the form F(ax + by, y' x, y' x ' x ) = Equations of the form F(x,xy' x -y,y' x ' x ) = Homogeneous equations Generalized homogeneous equations Equations invariant under scaling translation transformations Exact second-order equations Reduction of quasilinear equations to the normal form Methods of Regular Series Expansions with Respect to the Independent Variable or Small Parameter Method of expansion in powers of the independent variable Method of regular (direct) expansion in powers of the small parameter Pade approximants Perturbation Methods of Mechanics and Physics Preliminary remarks. A summary table of basic methods The method of scaled parameters (Lindstedt-Poincare method) Averaging method (Van der Pol-Krylov-Bogolyubov scheme) Method of two-scale expansions (Cole-Kevorkian scheme) Method of matched asymptotic expansions Galerkin Method and Its Modifications (Projection Methods) General form of an approximate solution Galerkin method The Bubnov-Galerkin method, the moment method, and the least squares method Collocation method The method of partitioning the domain The least squared error method Iteration and Numerical Methods The method of successive approximations (Cauchy problem) The Runge-Kutta method (Cauchy problem) Shooting method (boundary value problems) Method of accelerated convergence in eigenvalue problems Linear Equations of Arbitrary Order Linear Equations with Constant Coefficients Homogeneous linear equations Nonhomogeneous linear equations Linear Equations with Variable Coefficients Homogeneous linear equations. Structure of the general solution Utilization of particular solutions for reducing the order of die original equation Wronskian determinant and Liouville formula 54

5 viii CONTENTS Nonhomogeneous linear equations. Construction of the general solution Asymptotic Solutions of Linear Equations Fourth-order linear equations Higher-order linear equations Nonlinear Equations of Arbitrary Order Structure of the General Solution. Cauchy Problem Equations solved for the highest derivative. General solution The Cauchy problem. The existence and uniqueness theorem Equations Admitting Reduction of Order Equations not containing y,y' x,..., y x explicitly Equations not containing x explicitly (autonomous equations) Equations of the form F{ax + by,y' x,...,y n x ' ) ) = Equations of the form F(x, xy' x - y, y' xx,..., y^) = 0 and its generalizations Homogeneous equations Generalized homogeneous equations Equations of the form F(e Xx y n, y'jy, y'^/y y^/y) = Equations of the form F(x n e Xy, xy' x, x z y' x ' x x n y n) x ) = Other equations A Method for Construction of Solvable Equations of General Form Description of the method, Examples Lie Group and Discrete-Group Methods Lie Group Method. Point Transformations Local one-parameter Lie group of transformations. Invariance condition Group analysis of second-order equations. Structure of an admissible operator Utilization of local groups for reducing the order of equations and their integration Contact Transformations. Backlund Transformations. Formal Operators. Factorization Principle Contact transformations Backlund transformations. Formal operators and nonlocal variables Factorization principle First Integrals (Conservation Laws) Discrete-Group Method. Point Transformations Discrete-Group Method. The Method of RF-Pairs First-Order Differential Equations Simplest Equations with Arbitrary Functions Integrable in Closed Form Equations of the Form y' x = f(x) Equations of the Form y' x = f(y) Separable Equations y' x = f(x)g(y) Linear Equation g(x)y' x = fi(x)y + f o (x) Bernoulli Equation g(x)y' x = fi(x)y + f n (x)y n Homogeneous Equation y' x = f(y/x) 82

6 ix 1.2. Riccati Equation g(x)y' x = f2(x)y 2 + fi(x)y + fo(x) Preliminary Remarks Equations Containing Power Functions ; Equations of the form g(x)y' x = f 2 (x)y 2 + f o (x) Other equations Equations Containing Exponential Functions Equations with exponential functions Equations with power and exponential functions Equations Containing Hyperbolic Functions Equations widi hyperbolic sine and cosine Equations widi hyperbolic tangent and cotangent Equations Containing Logarithmic Functions Equations of the form g(x)y' x = fi(x)y 2 + f o (x) Equations of the form g(x)y' x = fi(x)y 2 + f\(x)y' x + f o (x) Equations Containing Trigonometric Functions Equations with sine Equations with cosine Equations with tangent Equations with cotangent Equations containing combinations of trigonometric functions Equations Containing Inverse Trigonometric Functions Equations containing arcsine Equations containing arccosine Equations containing arctangent Equations containing arccotangent Equations with Arbitrary Functions Equations containing arbitrary functions (but not containing their derivatives) Equations containing arbitrary functions and their derivatives Some Transformations Abel Equations of the Second Kind Equations of the Form yy' x -y = f{x) Preliminary remarks. Classification tables Solvable equations and their solutions Equations of the Form yy' x = f(x)y Equations of the Form yy' x = f\(x)y + f o (x) Preliminary remarks Solvable equations and their solutions Equations of the Form [g\(x)y + go(x)]y' x = f 2 (x)y 2 + fi(x)y + f o (x) Preliminary remarks Solvable equations and their solutions Some Types of First- and Second-Order Equations Reducible to Abel Equations of the Second Kind Quasi-homogeneous equations Equations of the theory of chemical reactors and the combustion theory Equations of the theory of nonlinear oscillations Second-order homogeneous equations of various types Second-order equations invariant under some transformations 137

7 1.4. Equations Containing Polynomial Functions of y Abel Equations of the First Kind y' x = f 3 (x)y 3 + h(x)y 2 + fi(x)y + f o (x) Preliminary remarks Solvable equations and their solutions EquationsoftheForm(A22y 2 +An%y+Aux 2 +Ao)y x =B 2 2y 2 +Bi2xy+B n x 2 +Bo Preliminary remarks. Some transformations Solvable equations and their solutions Equations of the Form (A222/ 2 + Mixy + A\\x 2 + A 2 y + A x x)y' x = B22I) 1 + B n xy + B n x 2 + B 2 y + B x x Preliminary remarks Solvable equations and their solutions Equations of the Form (A222/ 2 + A\ 2 xy + A\\x 2 + A2y + A\x + Ao)y' x = B 2 2y 2 + B n xy + B n x 2 + B 2 y + B x x + B Preliminary remarks. Some transformations Solvable equations and their solutions Equations of the Form (A 3 y 3 + A2xy 2 + A\x z y + AQX 3 + a\y + aox)y' x = B 3 y 3 + B 2 xy 2 + Bix 2 y + B 0 x 3 +b x y + b o x Equations of the Form f(x, y)y' x = g(x, y) Containing Arbitrary Parameters Equations Containing Power Functions Equations of the form y' x = f(x,y) Other equations Equations Containing Exponential Functions Equations with exponential functions Equations with power and exponential functions Equations Containing Hyperbolic Functions Equations Containing Logarithmic Functions Equations Containing Trigonometric Functions Equations Containing Combinations of Exponential, Hyperbolic, Logarithmic, and Trigonometric Functions Equations of the Form F(x,y,y' x ) = 0 Containing Arbitrary Parameters Equations of the Second Degree in y' x L Equations of the form f(x,y)(y' x ) 2 = g(x,y) Equations of the form f(x, y)(y' x ) 2 = g(x, y)y' x + h(x, y) Equations of the Third Degree in y' x Equations of the form f(x, y)(y' x ) 3 = g(x, y)y' x + h(x, y) Equations of the form f(x, y)(y' x ) 3 = g(x, y)(y' x ) 2 + h(x, y)y' x + r(x, y) Equations of the Form (y' x ) k = f(y) + g(x) Some transformations Classification tables and exact solutions Other Equations Equations containing algebraic and power functions with respect to y' x Equations containing exponential, logarithmic, and other functions with respect to y' x Equations of the Form f(x, y)y' x = g(x, y) Containing Arbitrary Functions Equations Containing Power Functions Equations Containing Exponential and Hyperbolic Functions Equations Containing Logarithmic Functions Equations Containing Trigonometric Functions Equations Containing Combinations of Exponential, Logarithmic, and Trigonometric Functions 201

8 xi 1.8. Equations of the Form F(x, y, y' x ) = 0 Containing Arbitrary Functions Some Equations Arguments of arbitrary functions depend on x and y Argument of arbitrary functions is y' x Arguments of arbitrary functions are linear witfi respect to y' x Arguments of arbitrary functions are nonlinear with respect to y' x Some Transformations Second-Order Differential Equations Linear Equations Representation of the General Solution Through a Particular Solution Equations Containing Power Functions Equations of the form y'^x + f(x)y = Equations of the form y'l x + j(x)y' x + g(x)y = Equations of the form (ax + b)y xx + f{x)y' x + g(x)y = Equations of the form x 2 y xx + f(x)y' x + g{x)y = Equations of the form (ax 2 +bx + c)y xx + f(x)y' x + g{x)y = Equations of the form (a,3x 3 + a2x 2 + aix + ao)y xx +f(x)y x +g(x)y = Q Equations of the form (a 4 x a x x + a o )y xx + f(x)y' x + g(x)y = Other equations Equations Containing Exponential Functions Equations with exponential functions Equations with power and exponential functions Equations Containing Hyperbolic Functions Equations with hyperbolic sine Equations with hyperbolic cosine Equations with hyperbolic tangent Equations with hyperbolic cotangent Equations containing combinations of hyperbolic functions Equations Containing Logariuimic Functions Equations of the form f(x)y xx + g(x)y = Equations of the form f(x)y xx + g(x)y' x + h(x)y = Equations Containing Trigonometric Functions Equations with sine Equations with cosine Equations with tangent Equations with cotangent Equations containing combinations of trigonometric functions Equations Containing Inverse Trigonometric Functions Equations with arcsine Equations with arccosine Equations with arctangent Equations with arccotangent Equations Containing Combinations of Exponential, Logarithmic, Trigonometric, and Other Functions Equations with Arbitrary Functions Equations containing arbitrary functions (but not containing their derivatives) Equations containing arbitrary functions and their derivatives Some Transformations 292

9 xii CONTENTS 2.2. Autonomous Equations y' xx = F{y, y' x ) Equations of the Form y% x -y' x = f(y) Equations of the Form y xx +f(y)y' x + y = Preliminary remarks Solvable equations and their solutions LienardEquations y' x ' x + f(y)y' x +g(y) = Preliminary remarks Solvable equations and their solutions Rayleigh Equations y xx + f(y' x ) + g(y) = Preliminary remarks. Some transformations Solvable equations and their solutions Emden-Fowler Equation y' x ' x = Ax n y m Exact Solutions Preliminary remarks. Classification table Solvable equations and their solutions First Integrals (Conservation Laws) First integrals withfc = First integrals with k = First integrals with k = First integrals with k = Some Formulas and Transformations Equations of the Form y' x ' x = Aix ni y mi + A 2 x ni y m Classification Table Exact Solutions Generalized Emden-Fowler Equation y xx = Ax n y m (y' x ) Classification Table Exact Solutions Some Formulas and Transformations A particular solution Discrete transformations of die generalized Emden-Fowler equation Reduction of the generalized Emden-Fowler equation to an Abel equation Equations of the Form y' x ' x = Aix nr y mi (y' x ) h + A 2 x n2 y mi (y' x ) h Modified Emden-Fowler Equation y' x ' x = Aix^y^ + A 2 x n y m Preliminary remarks. Classification table Solvable equations and their solutions Equations of the Form y' x ' x = (Aix ni y m ' +A2X n2 y mi )(y' x ) Classification table Solvable equations and their solutions Equations of the Form y x ' x = aax n y m (y'j + Ax n - l y m+1 (y' x ) Classification table Solvable equations and their solutions Other Equations (l x *l 2 ) Classification table Solvable equations and their solutions Equations of the Form y' x ' x - f(x)g(y)h(y' x ) Equations of the Form y' x ' x = f(x)g(y) Equations Containing Power Functions (h const) Equations Containing Exponential Functions (h const) 418

10 xiii Preliminary remarks Solvable equations and their solutions Equations Containing Hyperbolic Functions (h const) Equations Containing Trigonometric Functions (h const) Some Transformations Some Nonlinear Equations with Arbitrary Parameters Equations Containing Power Functions Equations of the form f(x, y)y xx + g(x, y) = l Equations of the form f(x,y)y' x x +g(x,y)y' x + h(x,y) = O Equations of the form f(x,y)y x ' x +g(x,y)(y x ) 2 + h(x,y)y' x +r(x,y) = Other equations Painleve Transcendents Preliminary remarks. Singular points of solutions First Painleve transcendent Second Painleve transcendent Third Painleve transcendent Fourth Painleve transcendent Fifth Painleve transcendent Sixth Painleve transcendent Equations Containing Exponential Functions Equations of the form f(x, y)y' x ' x + g(x, y) = Equations of the form f(x, y)y' x ' x + g(x, y)y' x + h(x, y) = l Equations of the form f(x,y)y' x x +g(x,y)(y l x) 2 + h(x,y)y x +r(x,y) = Other equations Equations Containing Hyperbolic Functions Equations with hyperbolic sine Equations with hyperbolic cosine Equations with hyperbolic tangent Equations widi hyperbolic cotangent Equations containing combinations of hyperbolic functions Equations Containing Logarithmic Functions Equations of me form f(x, y)y xx + g(x, y)y' x + h(x, y) = Other equations Equations Containing Trigonometric Functions Equations with sine Equations with cosine Equations with tangent Equations with cotangent Equations containing combinations of trigonometric functions Equations Containing the Combinations of Exponential, Hyperbolic, Logarithmic, and Trigonometric Functions Equations Containing Arbitrary Functions Equations of the Form F(x, y)y' xx + G(x,.y) = Arguments of arbitrary functions are algebraic and power functions of x and y Arguments of the arbitrary functions are other functions Equations of the Form F(x, y)y xx + G(x, y)y' x + H(x, y) = Argument of the arbitrary functions is a; Argument of the arbitrary functions is y Other arguments of the arbitrary functions 469

11 xiv CONTENTS M Equations of the Form F(x, y)y' x ' x + G m (x, y)(y' x ) m = 0 (M = 2, 3, 4) m= Argument of the arbitrary functions is x Argument of the arbitrary functions is y Other arguments of arbitrary functions Equations of the Form F(x, y, y' x )y xx + G(x, y,y' x ) = Arguments of the arbitrary functions depend on x or y Arguments of die arbitrary functions depend on x and y Arguments of die arbitrary functions depend on x, y, and y' x Equations Not Solved for Second Derivative Equations of General Form Equations containing arbitrary functions of two variables Equations containing arbitrary functions of three variables Some Transformations Third-Order Differential Equations Linear Equations Preliminary Remarks Equations Containing Power Functions Equations of the form / 3 (x)2/^x + f o (x)y = g(x) Equations of the form f 3 (x)y' x " xx + fi(x)y' x + f o (x)y = g(x) Equations of the form f 3 (x)y' x " xx + / 2 W * + /i (x)y' x + f o (x)y = g(x) Equations Containing Exponential Functions Equations with exponential functions Equations with power and exponential functions Equations Containing Hyperbolic Functions Equations widi hyperbolic sine Equations with hyperbolic cosine Equations with hyperbolic sine and cosine Equations with hyperbolic tangent Equations with hyperbolic cotangent Equations Containing Logarithmic Functions Equations with logarithmic functions Equations with power and logarithmic functions Equations Containing Trigonometric Functions Equations widi sine Equations widi cosine Equations with sine and cosine Equations with tangent Equations widi cotangent Equations Containing Inverse Trigonometric Functions Equations Containing Combinations of Exponential, Logarithmic, Trigonometric, and Otiier Functions Equations Containing Arbitrary Functions Equations of the form h(x)y' x " xx + fi(x)y' x + f o (x)y = g(x) Equations of the form / 3 (a;)c + h(x)y' x ' x + fi(x)y' x + f o (x)y = g(x) Equations of the Form y x " xx = Ax a y 0 (.y x y(y x ' x ) s Classification Table Equations of the Form y'j.'^ = Ay 0 566

12 XV Equations of the Form y^xx = Ax a y Equations with hi + \6\ * Some Transformations Equations of the Form y' x " xx = Hy)g(y' x My' x ' x ) Equations Containing Power Functions Equations Containing Exponential Functions Other Equations Nonlinear Equations with Arbitrary Parameters Equations Containing Power Functions Equations of die form f(x,y)y' xxx = g(x,y) Equations of the form y' x " xx = f(x, y,y' x ) Equations of the form f(x, y, y' x )y' x " xx + g(x, y, y x )y' x ' x + h(x, y, y' x ) = Otiier equations Equations Containing Exponential Functions Equations of the form y' x " xx = fix, y,y' x ) Other equations Equations Containing Hyperbolic Functions Equations with hyperbolic sine Equations with hyperbolic cosine Equations with hyperbolic tangent Equations with hyperbolic cotangent Equations Containing Logarithmic Functions Equations of the form y' x " xx = f(x, y, y' x ) Other equations Equations Containing Trigonometric Functions Equations widi sine Equations with cosine Equations with tangent Equations widi cotangent Nonlinear Equations Containing Arbitrary Functions Equations of the Form F(x,y)y' x " xx + G(x,y) = Arguments of the arbitrary functions are x or y Arguments of die arbitrary functions depend on x and y Equations of the Form F(x, y, y x )y xxx + G(x, y,y' x ) = Arguments of die arbitrary functions depend on x and y Arguments of the arbitrary functions depend on x, y, and y' x Equations of the Form F(x, y, y' x )y x " xx + G(x, y, y' x )y' x ' x + H(x, y,y' x ) = The arbitrary functions depend on x or y Arguments of arbitrary functions depend on x and y Arguments of arbitrary functions depend on x, y, and y' x Equations of the Form F(x, y, y'jy'^x + G a (x, y, y'm x ) a =0 634 a Arbitrary functions depend on x or y Arguments of arbitrary functions depend on x, y, and y' x Other Equations Equations of the form F(x, y, y' x, y'l)y'x"xx+g(x,y,y l x,y' x ' x ) = Q Equations of the form F(x, y, y' x, y xx,y' x " xx ) = 0 638

13 xvi CONTENTS 4. Fourth-Order Differential Equations Linear Equations Preliminary Remarks Equations Containing Power Functions Equations of the form f4(x)y' x " xxx + fo(x)y = g(x) Equations of the form f4ix)y' x " x ' xx + /i (x)y' x + f o (x)y = g(x) 642 l Equations of the form f4(x)y' x " x xx + f 2 (x)y' x ' x + f l (x)y' x + Mx)y = g(x) Other equations Equations Containing Exponential and Hyperbolic Functions Equations with exponential functions Equations widi hyperbolic functions Equations Containing Logarithmic Functions Equations Containing Trigonometric Functions Equations with sine and cosine Equations with tangent and cotangent Equations Containing Arbitrary Functions Equations of the form Mx)y xxxx + fi(x)y' x + fo(x)y = g(x) Equations of the form f4(x)y' x " x ' xx + f 1 ix)y l x' x + /i (x)y' x + f Q (x)y = g(x) Otiier equations Nonlinear Equations Equations Containing Power Functions Equations of the form y ' ^ = f(x,y) Equations of the form y^' xx = f(x, y,y' x ) Equations of the form y ^ = fix, y,y' x,y xx ) Equations of the form y^xx = fix, y, y' x, y'^, y' x ) Equations Containing Exponential Functions Equations of the form y xxxx = fix, y) Other equations Equations Containing Hyperbolic Functions Equations with hyperbolic sine Equations with hyperbolic cosine Equations with hyperbolic tangent Equations with hyperbolic cotangent Equations Containing Logaritiimic Functions Equations of the form y'^xx = f(x,y) Other equations Equations Containing Trigonometric Functions Equations with sine Equations widi cosine Equations widi tangent Equations with cotangent Equations Containing Arbitrary Functions Equations of the form y'j.'^ = fix,y) Equations of the form y^' xx = fix, y,y' x ) Equations of the form y'^xx = f(x,y,y' x,y'j. x ) Equations of the form y'^xx = f(x, y, y' x, y'^, y' x " xx ) Other equations 686

14 xvii 5. Higher-Order Differential Equations Linear Equations Preliminary Remarks Equations Containing Power Functions 689 n) Equations of the form f n ix)y x + fo(x)y = g(x) 689 n) Equations of the form f n (x)y x + f\(x)y' x + fo(x)y = g(x) Other equations Equations Containing Exponential and Hyperbolic Functions Equations widi exponential functions Equations widi hyperbolic functions Equations Containing Logaridimic Functions Equations Containing Trigonometric Functions Equations widi sine and cosine Equations with tangent and cotangent Equations Containing Arbitrary Functions 701 n) Equations of the form f n ix)y x + f\ix)y' x + foix)y = gix) Other equations Nonlinear Equations Equations Containing Power Functions Fifdi- and sixdi-order equations 705 n) Equations of the form y x = fix, y) 706 n) Equations of the form y x = fix, y, y' x, y x ' x ) Other equations Equations Containing Exponential Functions Fifth- and sixth-order equations 711 n) Equations of the form y x = fix, y) Otiier equations Equations Containing Hyperbolic Functions Equations with hyperbolic sine Equations with hyperbolic cosine Equations with hyperbolic tangent Equations widi hyperbolic cotangent Equations Containing Logaridimic Functions Equations of the form y^ = f(x, y) Other equations Equations Containing Trigonometric Functions Equations with sine Equations with cosine Equations with tangent Equations with cotangent Equations Containing Arbitrary Functions Fifth- and sixth-order equations 722 n) Equations of the form y x = fix, y) Equations of the form y<. n) = fix, y,y' x ) Equations of the form 2/ n) x = fix, y, y' x, y xx ) 727 n) Equations of the form fix,y)y x + g(x,y,y' x )y n x - 1) = fc(z,2/,2/ x,...,^- 2 >) Equations of the form y n) x = / (x, y, y' x,..., 2/ v " 1) ) Equations of the general form F (a;, j/,2/ x,...,2/ n) x ) =0 731

15 xviii CONTENTS Supplements Elementary Functions and Their Properties Trigonometric Functions 735 S.I.1-1. Simplest relations 735 S.I.1-2. Relations between trigonometric functions of single argument 735 S.I.1-3. Reduction formulas 735 S.I Addition and subtraction of trigonometric functions 736 S.I.1-5. Products of trigonometric functions 736 S.I.1-6. Powers of trigonometric functions 736 S.I.1-7. Addition formulas 737 S.I.1-8. Trigonometric functions of multiple arguments 737 S.I.1-9. Trigonometric functions of half argument 737 S Euler and de Moivre formulas. Relationship with hyperbolic functions 737 S.I Differentiation formulas 737 S.I Expansion into power series Hyperbolic Functions 738 S.l.2-1. Definitions 738 S.l.2-2. Simplest relations 738 S Relations between hyperbolic functions of single argument (x > 0) S.l.2-4. Addition formulas 738 S Addition and subtraction of hyperbolic functions 739 S Products of hyperbolic functions 739 S Powers of hyperbolic functions 739 S Hyperbolic functions of multiple arguments 739 S.I.2-9. Relationship with trigonometric functions 740 S Differentiation formulas 740 S Expansion into power series Inverse Trigonometric Functions 740 S Definitions and some properties 740 S.l.3-2. Simplest formulas 741 S Relations between inverse trigonometric functions 741 S Addition and subtraction of inverse trigonometric functions 741 S.l.3-5. Differentiation formulas 741 S Expansion into power series Inverse Hyperbolic Functions 742 S Relationships with logaridimic functions 742 S Relations between inverse hyperbolic functions 742 S Addition and subtraction of inverse hyperbolic functions 742 S Differentiation formulas 742 S Expansion into power series Special Functions and Their Properties Some Symbols and Coefficients 743 S Factorials 743 S Binomial coefficients 743 S Pochhammer symbol Error Functions and Exponential Integral 744 S Error function and complementary error function 744 S Exponential integral 744 S Logarithmic integral 745

16 xix Gamma and Beta Functions 745 S Gamma function 745 S Logarithmic derivative of the gamma function 746 S Beta function Incomplete Gamma and Beta Functions 747 S Incomplete gamma function 747 S Incomplete beta function Bessel Functions 748 S Definitions and basic formulas 748 S Bessel functions for v = ±n±\, where n = 0, 1, S Bessel functions for v = ±n, where n = 0, 1, 2, 749 S Wronskians and similar formulas 749 S Integral representations 749 S Asymptotic expansions 750 S Zeros and orthogonality properties of die Bessel functions 750 S Hankel functions (Bessel functions of the diird kind) Modified Bessel Functions 751 S Definitions. Basic formulas 751 S Modified Bessel functions for v = ±n±\, where n = 0, 1, 2, 751 S Modified Bessel functions for v = n, where n = 0, 1, 2, 752 S Wronskians and similar formulas 752 S Integral representations 752 S Asymptotic expansions as x > oo Degenerate Hypergeometric Functions 753 S Definitions. The Kummer's series 753 S Some transformations and linear relations 753 S Differentiation formulas and Wronskian 753 S Degenerate hypergeometric functions for n = 0, 1, 2, 754 S Integral representations 754 S Asymptotic expansion as \x\ -> oo 754 S Whittaker functions Hypergeometric Functions 755 S Definition. The hypergeometric series 755 S Basic properties 755 S Integral representations Legendre Functions and Legendre Polynomials 756 S Definitions. Basic formulas 756 S Trigonometric expansions 756 S Some relations 757 S Integral representations 757 S Legendre polynomials 757 S Zeros of the Legendre polynomials and die generating function 758 S Associated Legendre functions Parabolic Cylinder Functions 758 S Definitions. Basic formulas 758 S Integral representations 759 S Asymptotic expansion as \z\ -* oo Orthogonal Polynomials 759 S Laguerre polynomials and generalized Laguerre polynomials 759 S Chebyshev polynomials 760 S Hermite polynomials 761

17 xx CONTENTS S Gegenbauer polynomials 762 S Jacobi polynomials 762 S The Weierstrass Function 762 S Definitions 762 S Some properties 762 S.3. Tables of Indefinite Integrals Integrals Containing Rational Functions 763 S Integrals containing a + bx 763 S Integrals containing a + x and b + x 763 S Integrals containing a 2 + x S Integrals containing a 2 -x S Integrals containing a 3 +x S Integrals containing a 3 - x S Integrals containing a 4 + a; Integrals Containing Irrational Functions 767 S Integrals containing x x l S Integrals containing (a + bxfl S Integrals containing (a: 2 + a 2 ) 1 / S Integrals containing (a; 2 - a 2 ) 1 / S Integrals containing (a 2 - x 2 ) 1 / S Reduction formulas Integrals Containing Exponential Functions Integrals Containing Hyperbolic Functions 769 S Integrals containing cosha; 769 S Integrals containing sinhx 770 S Integrals containing tanhx or cotha; Integrals Containing Logarithmic Functions Integrals Containing Trigonometric Functions 773 S Integrals containing cos x 773 S Integrals containing sin x 774 S Integrals containing sin a; and cos a; 776 S Reduction formulas 776 S Integrals containing tana; and cota; Integrals Containing Inverse Trigonometric Functions 777 References 779 Index * 783