Handbook of Exact Solutions to Mathematical Equations

1         Algebraic and Transcendental Equations                                                           

1.1.       Algebraic Equations  

1.1.1.         Linearand Quadratic Equations

1.1.2.         Cubic Equations 

1.1.3.         Equations of the Fourth Degree

1.1.4.         Equations of the Fifth Degree

1.1.5.         Algebraic Equations of Arbitrary Degree

1.1.6.         Systems of Linear Algebraic Equations

1.2.       Trigonometric Equations

1.2.1.         Binomial Trigonometric Equations

1.2.2.         Trigonometric Equations Containing Several Terms

1.2.3.         Trigonometric Equations of the General Form

1.3.       Other Transcendental Equations

1.3.1.         Equations Containing Exponential Functions

1.3.2.         Equations Containing Hyperbolic Functions

1.3.3.         Equations Containing Logarithmic Functions

References for Chapter 1

2         Ordinary Differential Equations                                                                       

2.1.       First-Order Ordinary Differential Equations

2.1.1.         Simplest First-Order ODEs

2.1.2.         Riccati Equations

2.1.3.         Abel Equations

2.1.4.         Other First-Order ODEs Solved for the Derivative

2.1.5.         ODEs Not Solved for the Derivative and ODEs Defined Parametrically

2.2.       Second-Order Linear Ordinary Differential Equations

2.2.1.         Preliminary Remarks and Some Formulas

2.2.2.         Equations Involving Power Functions

2.2.3.         Equations Involving Exponential and Other Elementary Functions    

2.2.4.         Equations Involving Arbitrary Functions

2.3.       Second-Order Nonlinear Ordinary Differential Equations

2.3.1.         Equations of the Form yx′′x = f (x, y)

2.3.2.         Equations of the Form f (x, y)yx′′x = g(x, y, yx′ )

2.3.3.         ODEs of General Form Containing Arbitrary Functions of Two Arguments

2.4.       Higher-Order Ordinary Differential Equations

2.4.1.         Higher-Order Linear Ordinary Differential Equations

2.4.2.         Third-andFourth-OrderNonlinearOrdinaryDifferentialEquations

2.4.3.         Higher-Order Nonlinear Ordinary Differential Equations

References for Chapter 2

3         Systems of Ordinary Differential Equations                                                   

3.1.       Linear Systems of ODEs

3.1.1.         Systems of Two First-Order ODEs

3.1.2.         Systems of Two Second-Order ODEs

3.1.3.         Other Systems of Two ODEs

3.1.4.         Systems of Three and More ODEs

3.2.       Nonlinear Systems of Two ODEs

3.2.1.         Systems of First-Order ODEs

3.2.2.         Systems of Second- and Third-Order ODEs

3.3.       Nonlinear Systems of Three or More ODEs

3.3.1.         Systems of Three ODEs

3.3.2.         Equations of Dynamics of a Rigid Body with a Fixed Point

References for Chapter 3

4         First-Order Partial Differential Equations                                                     

4.1.       Linear Partial Differential Equations in Two Independent Variables

4.1.1.         Preliminary Remarks. Solution Methods

4.1.2.         Equations of the Form f (x, y)u_x + g(x, y)u_y = 0

4.1.3.         Equations of the Form f (x, y)u_x + g(x, y)u_y = h(x, y)

4.1.4.         Equations of the Form f (x, y)u_x + g(x, y)u_y = h(x, y)u + r(x, y)

4.2.       Quasilinear Partial Differential Equations in Two Independent Variables     

4.2.1.         Preliminary Remarks. Solution Methods

4.2.2.         Equations of the Form f (x, y)u_x + g(x, y)u_y = h(x, y, u)

4.2.3.         Equations of the Form u_x + f (x, y, u)u_y = 0

4.2.4.         Equations of the Form u_x + f (x, y, u)u_y = g(x, y, u)

4.3.       NonlinearPartialDifferentialEquationsinTwoIndependent Variables

4.3.1.         Preliminary Remarks. A Complete Integral

4.3.2.         Equations Quadratic in One Derivative

4.3.3.         Equations Quadratic in Two Derivatives

4.3.4.         Equations with Arbitrary Nonlinearities in Derivatives

References for Chapter 4

 

5         Linear Equations and Problems of Mathematical Physics                                  

5.1.       Parabolic Equations

5.1.1.         Heat (Diffusion) Equation u_t = au_xx

5.1.2.         Nonhomogeneous Heat Equation u_t = au_xx + Φ(x, t)

5.1.3.         Heat Type Equation of the Form u_t = au_xx + bu_x + cu + Φ(x, t)

5.1.4.         Heat Equation with Axial Symmetry u_t = a(u_rr + r⁻¹u_r)

5.1.5.         Nonhomogeneous Heat Equation with Axial Symmetry

u_t = a(u_rr + r⁻¹u_r) + Φ(r, t)

5.1.6.         Heat Equation with Central Symmetry u_t = a(u_rr + 2r⁻¹u_r)      

5.1.7.         Nonhomogeneous Heat Equation with Central Symmetry u_t = a(u_rr + 2r⁻¹u_r) + Φ(r, t)

5.1.8.         Heat Type Equation of the Form u_t = u_xx + (1 − 2β)x⁻¹u_x

5.1.9.         Heat Type Equation of the Form u_t = [f (x)u_x]_x

5.1.10.   Equations of the Form s(x)u_t = [p(x)u_x]_xq(x)u + Φ(x, t)

5.1.11.   Liquid-Film Mass Transfer Equation (1y²)u_x = au_yy

5.1.12.    Equations of the Diffusion (Thermal) Boundary Layer n2

5.1.13.   t2mxx Schro¨dinger Equation inu = −     u     + U (x)u

5.2.       Hyperbolic Equations

5.2.1.         Wave Equation u_tt = a²u_xx

5.2.2.         Nonhomogeneous Wave Equation u_tt = a²u_xx + Φ(x, t)

5.2.3.        Klein–Gordon Equation u_tt = a²u_xx       bu

5.2.4.         Nonhomogeneous Klein–Gordon Equation u_tt = a²u_xx         bu + Φ(x, t)

5.2.5.         Wave Equation with Axial Symmetry u_tt = a²(u_rr + r⁻¹u_r) + Φ(r, t)

5.2.6.         Wave Equation with Central Symmetry u_tt = a²(u_rr + 2r⁻¹u_r) + Φ(r, t)

5.2.7.        Equations of the Form s(x)u_tt = [p(x)u_x]_x       q(x)u + Φ(x, t)

5.2.8.         Telegraph Type Equations u_tt + ku_t = a²u_xx + bu_x + cu + Φ(x, t)

5.3.       Elliptic Equations

5.3.1.         Laplace Equation ∆u = 0

5.3.2.         Poisson Equation ∆u + Φ(x, y) = 0

5.3.3.        Helmholtz Equation ∆u + λu =      Φ(x, y)

5.3.4.         Convective Heat and Mass Transfer Equations

5.3.5.         Equations of Heat and Mass Transfer in Anisotropic Media

5.3.6.         Tricomi and Related Equations

5.4.       Simplifications of Second-Order Linear Partial Differential Equations   

5.4.1.         Reduction of PDEs in Two Independent Variables to Canonical Forms

5.4.2.         Simplifications of Linear Constant-Coefficient Partial Differential Equations

5.5.       Third-Order Linear Partial Differential Equations

5.5.1.         Equations Containing the First Derivative in t and the Third Derivative in x

5.5.2.         Equations Containing the First Derivative in t and a Mixed Third Derivative

5.5.3.         Equations Containing the Second Derivative in t and a Mixed Third Derivative                                                                                    

      5.6.  Fourth-Order Linear Partial Differential Equations                                   

      5.6.1. Equation of Transverse Vibration of an Elastic Rod u_tt + a²u_xxxx = 0                                                                                       

      5.6.2. Nonhomogeneous Equation of the Form u_tt + a²u_xxxx = Φ(x, t)              

      5.6.3. Biharmonic Equation ∆∆u = 0                                                               

      5.6.4. Nonhomogeneous Biharmonic Equation ∆∆u = Φ(x, y)                         

      References for Chapter 5                                                                                    

6   Nonlinear Equations of Mathematical Physics                                               

      6.1. Parabolic Equations                                                                                   

      6.1.1. Quasilinear Heat Equations with a Source of the Form u_t = au_xx + f (u) 

      6.1.2. Burgers Type Equations and Related PDEs                                              

      6.1.3. Reaction-Diffusion Equations of the Form u_t = [f (u)u_x]_x + g(u)             

      6.1.4. Other Reaction-Diffusion and Heat PDEs with Variable Transfer Coefficient                                                                  

      6.1.5. Convection-Diffusion Type PDEs                                                            

      6.1.6. NonlinearSchro¨dinger EquationsandRelatedPDEs

      6.2. Hyperbolic Equations                                                                               

      6.2.1. Nonlinear Klein–Gordon Equations of the Form u_tt = au_xx + f (u)            

      6.2.2. OtherNonlinearWaveTypeEquations

      6.3. Elliptic Equations                                                                                    

      6.3.1. Heat Equations with Nonlinear Source of the Form u_xx + u_yy = f (u)

      6.3.2. Stationary Anisotropic Heat/Diffusion Equations of the Form [f (x)u_x]_x + [g(y)u_y]_y = h(u)

      6.3.3. Stationary Anisotropic Heat/Diffusion Equations of the Form               

      [f (u)u_x]_x + [g(u)u_y]_y = h(u)

      6.4. Other Second-Order Equations                                                                 

      6.4.1. EquationsofTransonicGasFlow

      6.4.2. Monge–Ampe`reTypeEquations

      6.5. Higher-Order Equations

      6.5.1. Third-OrderEquations

      6.5.2. Fourth-OrderEquations

      References for Chapter 6                                                                                   

7   Systems of Partial Differential Equations                                                      

      7.1. Systems of Two First-Order PDEs                                                            

      7.1.1. LinearSystemsofTwoFirst-OrderPDEs

      7.1.2. Nonlinear Systems of the Form u_x = F (u, w), w_t = G(u, w)

      7.1.3. Gas Dynamic Type Systems Linearizable with the Hodograph               

      Transformation

      7.2.  Systems of Two Second-Order PDEs 

        7.2.1. LinearSystemsofTwoSecond-OrderPDEs

7.2.2.         Nonlinear Parabolic Systems of the Form

u_t = au_xx + F (u, w), w_t = bw_xx + G(u, w)

7.2.3.         Nonlinear Parabolic Systems of the Form

u_t = ax⁻ⁿ(xⁿu_x)_x + F (u, w), w_t = bx⁻ⁿ(xⁿw_x)_x + G(u, w)

7.2.4.         Nonlinear Hyperbolic Systems of the Form

u_tt = au_xx + F (u, w), w_tt = bw_xx + G(u, w)

7.2.5.         Nonlinear Hyperbolic Systems of the Form

u_tt = ax⁻ⁿ(xⁿu_x)_x + F (u, w), w_tt = bx⁻ⁿ(xⁿw_x)_x + G(u, w)              

7.2.6.         Nonlinear Elliptic Systems of the Form

∆u = F (u, w), ∆w = G(u, w)

7.3.       PDE Systems of General Form

7.3.1.         Linear Systems

7.3.2.         Nonlinear Systems of Two Equations Involving the First Derivatives with Respect to t

7.3.3.         Nonlinear Systems of Two Equations Involving the Second Derivatives with Respect to t

References for Chapter 7

8         Integral Equations                                                                                          

8.1.       IntegralEquationsoftheFirstKindwithVariableLimitofIntegration

8.1.1.         Linear Volterra Integral Equations of the First Kind

8.1.2.         Nonlinear Volterra Integral Equations of the First Kind

8.2.       Integral Equations of the Second Kind with Variable Limit of Integration   

8.2.1.         Linear Volterra Integral Equations of the Second Kind

8.2.2.         Nonlinear Volterra Integral Equations of the Second Kind

8.3.       Equations of the First Kind with Constant Limits of Integration

8.3.1.         Linear Fredholm Integral Equations of the First Kind

8.3.2.         Nonlinear Fredholm Integral Equations of the First Kind

8.4.       Equations of the Second Kind with Constant Limits of Integration

8.4.1.         Linear Fredholm Integral Equations of the Second Kind

8.4.2.         Nonlinear Fredholm Integral Equations of the Second Kind

References for Chapter 8

9         Difference and Functional Equations                                                             

9.1.       Difference Equations

9.1.1.         Difference Equations with Discrete Argument

9.1.2.         Difference Equations with Continuous Argument

9.2.       Linear Functional Equations in One Independent Variable

9.2.1.         Linear Functional Equations Involving Unknown Function with

Two Different Arguments

9.2.2.         Other Linear Functional Equations

9.3.       Nonlinear Functional Equations in One Independent Variable

9.3.1.         Functional Equations with Quadratic Nonlinearity

9.3.2.         Functional Equations with Power Nonlinearity

9.3.3.         Nonlinear Functional Equation of General Form

9.4.       Functional Equations in Several Independent Variables

9.4.1.         Linear Functional Equations

9.4.2.         Nonlinear Functional Equations

References for Chapter 9

10     Ordinary Functional Differential Equations                                                   

10.1.   First-Order Linear Ordinary Functional Differential Equations

10.1.1.    ODEs with Constant Delays

10.1.2.    Pantograph-Type ODEs with Proportional Arguments

10.1.3.    Other Ordinary Functional Differential Equations

10.2.   First-Order Nonlinear Ordinary Functional Differential Equations

10.2.1.    ODEs with Constant Delays

10.2.2.    Pantograph-Type ODEs with Proportional Arguments

10.2.3.    Other Ordinary Functional Differential Equations

10.3.   Second-Order Linear Ordinary Functional Differential Equations

10.3.1.    ODEs with Constant Delays

10.3.2.    Pantograph-Type ODEs with Proportional Arguments

10.3.3.    Other Ordinary Functional Differential Equations

10.4.   Second-Order Nonlinear Ordinary Functional Differential Equations

10.4.1.    ODEs with Constant Delays

10.4.2.    Pantograph-Type ODEs with Proportional Arguments

10.4.3.    Other Ordinary Functional Differential Equations

10.5.   Higher-Order Ordinary Functional Differential Equations

10.5.1.    Linear Ordinary Functional Differential Equations

10.5.2.    Nonlinear Ordinary Functional Differential Equations

References for Chapter 10

11     Partial Functional Differential Equations                                                       

11.1.   Linear Partial Functional Differential Equations

11.1.1.    PDEs with Constant Delay

11.1.2.    PDEs with Proportional Delay

11.1.3.    PDEs with Anisotropic Time Delay

11.2.   Nonlinear PDEs with Constant Delays

11.2.1.    Parabolic Equations

11.2.2.    Hyperbolic Equations

11.3.   Nonlinear PDEs with Proportional Arguments

11.3.1.    Parabolic Equations

11.3.2.    Hyperbolic Equations

11.4.   Partial Functional Differential Equations with Arguments of General Type

11.4.1.    Parabolic Equations

11.4.2.    Hyperbolic Equations

11.5.   PDEs with Anisotropic Time Delay

11.5.1.    Parabolic Equations

11.5.2.    Hyperbolic Equations

References for Chapter 11

Biography

Andrei D. Polyanin, D.Sc., Ph.D., is a well-known scientist of broad interests and is active in various areas of mathematics, theory of heat and mass transfer, hydrodynamics, and chemical engineering sciences. He is one of the most prominent authors in the field of reference literature on mathematics. Professor Polyanin graduated with honors from the Department of Mechan- ics and Mathematics at the Lomonosov Moscow State University in 1974. Since 1975, Professor Polyanin has been working at the Ishlinsky Institute for Problems in Mechanics of the Russian (former USSR) Academy of Sciences, where he defended his Ph.D. in 1981 and D.Sc. degree in 1986.

Professor Polyanin has made important contributions to the theory of differential and integral equations, mathematical physics, applied and engineering mathematics, the theory of heat and mass transfer, and hydrodynamics. He develops analytical methods for constructing solutions to mathematical equations of various types and has obtained a huge number of exact solutions of ordinary differential, partial differential, delay partial differential, integral, and functional equations.

Professor Polyanin is an author of more than 30 books and over 270 articles and holds three patents. His books include V. F. Zaitsev and A. D. Polyanin, Discrete- Group Methods for Integrating Equations of Nonlinear Mechanics, CRC Press, 1994; A. D. Polyanin and V. V. Dilman, Methods of Modeling Equations and Analogies in Chemical Engineering, CRC Press/Begell House, Boca Raton, 1994; A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press, 1995 (2nd edition in 2003); A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, 1998 (2nd edition in 2008); A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002 (2nd edition, co-authored with V. E. Nazaikinskii, in 2016); A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, 2002; A. D. Polyanin, A. M. Kutepov, et al., Hydrodynamics, Mass and Heat Transfer in Chemical Engineering, Taylor & Francis, 2002; A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, 2004 (2nd edition in 2012); A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists, Chapman & Hall/CRC Press, 2007; A. D. Polyanin and V. F. Zaitsev, Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems, CRC Press, 2018; A. D. Polyanin and A. I. Zhurov, Separation of Variables and Exact Solutions to Nonlinear PDEs, CRC Press, 2022, and A. D. Polyanin, V. G. Sorokin, and A. I. Zhurov, Delay Ordinary and Partial Differential Equations, CRC Press, 2023.

Professor Polyanin is editor-in-chief of the international scientific educational website EqWorld— The World of Mathematical Equations and a member of the editorial boards of several journals.