Handbook of Exact Solutions to Mathematical Equations

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.

• 1 Algebraic and Transcendental Equations 1.1. Algebraic Equations 1.1.1. LinearandQuadraticEquations1.1.2. Cubic Equations 1.1.3. EquationsoftheFourthDegree1.1.4. EquationsoftheFifthDegree1.1.5. Algebraic Equations of Arbitrary Degree1.1.6. Systems of Linear Algebraic Equations1.2. Trigonometric Equations1.2.1. Binomial Trigonometric Equations1.2.2. Trigonometric Equations Containing Several Terms1.2.3. Trigonometric Equations of the General Form1.3. Other Transcendental Equations1.3.1. Equations Containing Exponential Functions1.3.2. Equations Containing Hyperbolic Functions1.3.3. Equations Containing Logarithmic FunctionsReferences for Chapter 12 Ordinary Differential Equations 2.1. First-Order Ordinary Differential Equations2.1.1. Simplest First-Order ODEs2.1.2. Riccati Equations2.1.3. Abel Equations2.1.4. Other First-Order ODEs Solved for the Derivative2.1.5. ODEs Not Solved for the Derivative and ODEs Defined Parametrically2.2. Second-Order Linear Ordinary Differential Equations2.2.1. Preliminary Remarks and Some Formulas2.2.2. Equations Involving Power Functions2.2.3. Equations Involving Exponential and Other Elementary Functions 2.2.4. Equations Involving Arbitrary Functions2.3. Second-Order Nonlinear Ordinary Differential Equations2.3.1. 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 Arguments2.4. Higher-Order Ordinary Differential Equations2.4.1. Higher-Order Linear Ordinary Differential Equations2.4.2. Third-andFourth-OrderNonlinearOrdinaryDifferentialEquations2.4.3. Higher-Order Nonlinear Ordinary Differential EquationsReferences for Chapter 23 Systems of Ordinary Differential Equations 3.1. Linear Systems of ODEs3.1.1. Systems of Two First-Order ODEs3.1.2. Systems of Two Second-Order ODEs3.1.3. Other Systems of Two ODEs3.1.4. Systems of Three and More ODEs3.2. Nonlinear Systems of Two ODEs3.2.1. Systems of First-Order ODEs3.2.2. Systems of Second- and Third-Order ODEs3.3. Nonlinear Systems of Three or More ODEs3.3.1. Systems of Three ODEs3.3.2. Equations of Dynamics of a Rigid Body with a Fixed PointReferences for Chapter 34 First-Order Partial Differential Equations 4.1. Linear Partial Differential Equations in Two Independent Variables4.1.1. Preliminary Remarks. Solution Methods4.1.2. Equations of the Form f (x, y)ux + g(x, y)uy = 04.1.3. Equations of the Form f (x, y)ux + g(x, y)uy = h(x, y)4.1.4. Equations of the Form f (x, y)ux + g(x, y)uy = h(x, y)u + r(x, y)4.2. Quasilinear Partial Differential Equations in Two Independent Variables 4.2.1. Preliminary Remarks. Solution Methods4.2.2. Equations of the Form f (x, y)ux + g(x, y)uy = h(x, y, u)4.2.3. Equations of the Form ux + f (x, y, u)uy = 04.2.4. Equations of the Form ux + f (x, y, u)uy = g(x, y, u)4.3. NonlinearPartialDifferentialEquationsinTwoIndependent Variables4.3.1. Preliminary Remarks. A Complete Integral4.3.2. Equations Quadratic in One Derivative4.3.3. Equations Quadratic in Two Derivatives4.3.4. Equations with Arbitrary Nonlinearities in DerivativesReferences for Chapter 45 Linear Equations and Problems of Mathematical Physics 5.1. Parabolic Equations5.1.1. Heat (Diffusion) Equation ut = auxx5.1.2. Nonhomogeneous Heat Equation ut = auxx + Φ(x, t)5.1.3. Heat Type Equation of the Form ut = auxx + bux + cu + Φ(x, t)5.1.4. Heat Equation with Axial Symmetry ut = a(urr + r−1ur)5.1.5. Nonhomogeneous Heat Equation with Axial Symmetryut = a(urr + r−1ur) + Φ(r, t)5.1.6. Heat Equation with Central Symmetry ut = a(urr + 2r−1ur) 5.1.7. Nonhomogeneous Heat Equation with Central Symmetryut = a(urr + 2r−1ur) + Φ(r, t)5.1.8. Heat Type Equation of the Form ut = uxx + (1 − 2β)x−1ux5.1.9. Heat Type Equation of the Form ut = [f (x)ux]x5.1.10. −Equations of the Form s(x)ut = [p(x)ux]x q(x)u + Φ(x, t)5.1.11. −Liquid-Film Mass Transfer Equation (1 y2)ux = auyy5.1.12. Equations of the Diffusion (Thermal) Boundary Layern25.1.13. t2mxxSchro¨dinger Equation inu = − u + U (x)u5.2. Hyperbolic Equations5.2.1. Wave Equation utt = a2uxx5.2.2. Nonhomogeneous Wave Equation utt = a2uxx + Φ(x, t)5.2.3. −Klein–Gordon Equation utt = a2uxx bu5.2.4. Nonhomogeneous Klein–Gordon Equation−utt = a2uxx bu + Φ(x, t)5.2.5. Wave Equation with Axial Symmetryutt = a2(urr + r−1ur) + Φ(r, t)5.2.6. Wave Equation with Central Symmetryutt = a2(urr + 2r−1ur) + Φ(r, t)5.2.7. −Equations of the Form s(x)utt = [p(x)ux]x q(x)u + Φ(x, t)5.2.8. Telegraph Type Equations utt + kut = a2uxx + bux + cu + Φ(x, t)5.3. Elliptic Equations5.3.1. Laplace Equation ∆u = 05.3.2. Poisson Equation ∆u + Φ(x, y) = 05.3.3. −Helmholtz Equation ∆u + λu = Φ(x, y)5.3.4. Convective Heat and Mass Transfer Equations5.3.5. Equations of Heat and Mass Transfer in Anisotropic Media5.3.6. Tricomi and Related Equations5.4. Simplifications of Second-Order Linear Partial Differential Equations 5.4.1. Reduction of PDEs in Two Independent Variables to Canonical Forms5.4.2. Simplifications of Linear Constant-Coefficient Partial Differential Equations5.5. Third-Order Linear Partial Differential Equations5.5.1. Equations Containing the First Derivative in t and the Third Derivative in x5.5.2. Equations Containing the First Derivative in t and a Mixed Third Derivative5.5.3. 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 utt + a2uxxxx = 0 5.6.2. Nonhomogeneous Equation of the Form utt + a2uxxxx = 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 ut = auxx + f (u) 6.1.2. Burgers Type Equations and Related PDEs 6.1.3. Reaction-Diffusion Equations of the Form ut = [f (u)ux]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 EquationsandRelatedPDEs6.2. Hyperbolic Equations 6.2.1. Nonlinear Klein–Gordon Equations of the Form utt = auxx + f (u) 6.2.2. OtherNonlinearWaveTypeEquations6.3. Elliptic Equations 6.3.1. Heat Equations with Nonlinear Source of the Form uxx + uyy = f (u)6.3.2. Stationary Anisotropic Heat/Diffusion Equations of the Form [f (x)ux]x + [g(y)uy]y = h(u)6.3.3. Stationary Anisotropic Heat/Diffusion Equations of the Form [f (u)ux]x + [g(u)uy]y = h(u)6.4. Other Second-Order Equations 6.4.1. EquationsofTransonicGasFlow6.4.2. Monge–Ampe`reTypeEquations6.5. Higher-Order Equations6.5.1. Third-OrderEquations6.5.2. Fourth-OrderEquationsReferences for Chapter 6 7 Systems of Partial Differential Equations 7.1. Systems of Two First-Order PDEs 7.1.1. LinearSystemsofTwoFirst-OrderPDEs7.1.2. Nonlinear Systems of the Form ux = F (u, w), wt = G(u, w)7.1.3. Gas Dynamic Type Systems Linearizable with the Hodograph Transformation7.2. Systems of Two Second-Order PDEs 7.2.1. LinearSystemsofTwoSecond-OrderPDEs7.2.2. Nonlinear Parabolic Systems of the Formut = auxx + F (u, w), wt = bwxx + G(u, w)7.2.3. Nonlinear Parabolic Systems of the Formut = ax−n(xnux)x + F (u, w), wt = bx−n(xnwx)x + G(u, w)7.2.4. Nonlinear Hyperbolic Systems of the Formutt = auxx + F (u, w), wtt = bwxx + G(u, w)7.2.5. Nonlinear Hyperbolic Systems of the Formutt = ax−n(xnux)x + F (u, w), wtt = bx−n(xnwx)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 Form7.3.1. Linear Systems7.3.2. Nonlinear Systems of Two Equations Involving the First Derivatives with Respect to t7.3.3. Nonlinear Systems of Two Equations Involving the Second Derivatives with Respect to tReferences for Chapter 78 Integral Equations 8.1. IntegralEquationsoftheFirstKindwithVariableLimitofIntegration8.1.1. Linear Volterra Integral Equations of the First Kind8.1.2. Nonlinear Volterra Integral Equations of the First Kind8.2. Integral Equations of the Second Kind with Variable Limit of Integration 8.2.1. Linear Volterra Integral Equations of the Second Kind8.2.2. Nonlinear Volterra Integral Equations of the Second Kind8.3. Equations of the First Kind with Constant Limits of Integration8.3.1. Linear Fredholm Integral Equations of the First Kind8.3.2. Nonlinear Fredholm Integral Equations of the First Kind8.4. Equations of the Second Kind with Constant Limits of Integration8.4.1. Linear Fredholm Integral Equations of the Second Kind8.4.2. Nonlinear Fredholm Integral Equations of the Second KindReferences for Chapter 89 Difference and Functional Equations 9.1. Difference Equations9.1.1. Difference Equations with Discrete Argument9.1.2. Difference Equations with Continuous Argument9.2. Linear Functional Equations in One Independent Variable9.2.1. Linear Functional Equations Involving Unknown Function withTwo Different Arguments9.2.2. Other Linear Functional Equations9.3. Nonlinear Functional Equations in One Independent Variable9.3.1. Functional Equations with Quadratic Nonlinearity9.3.2. Functional Equations with Power Nonlinearity9.3.3. Nonlinear Functional Equation of General Form9.4. Functional Equations in Several Independent Variables9.4.1. Linear Functional Equations9.4.2. Nonlinear Functional EquationsReferences for Chapter 910 Ordinary Functional Differential Equations 10.1. First-Order Linear Ordinary Functional Differential Equations10.1.1. ODEs with Constant Delays10.1.2. Pantograph-Type ODEs with Proportional Arguments10.1.3. Other Ordinary Functional Differential Equations10.2. First-Order Nonlinear Ordinary Functional Differential Equations10.2.1. ODEs with Constant Delays10.2.2. Pantograph-Type ODEs with Proportional Arguments10.2.3. Other Ordinary Functional Differential Equations10.3. Second-Order Linear Ordinary Functional Differential Equations10.3.1. ODEs with Constant Delays10.3.2. Pantograph-Type ODEs with Proportional Arguments10.3.3. Other Ordinary Functional Differential Equations10.4. Second-Order Nonlinear Ordinary Functional Differential Equations10.4.1. ODEs with Constant Delays10.4.2. Pantograph-Type ODEs with Proportional Arguments10.4.3. Other Ordinary Functional Differential Equations10.5. Higher-Order Ordinary Functional Differential Equations10.5.1. Linear Ordinary Functional Differential Equations10.5.2. Nonlinear Ordinary Functional Differential EquationsReferences for Chapter 1011 Partial Functional Differential Equations 11.1. Linear Partial Functional Differential Equations11.1.1. PDEs with Constant Delay11.1.2. PDEs with Proportional Delay11.1.3. PDEs with Anisotropic Time Delay11.2. Nonlinear PDEs with Constant Delays11.2.1. Parabolic Equations11.2.2. Hyperbolic Equations11.3. Nonlinear PDEs with Proportional Arguments11.3.1. Parabolic Equations11.3.2. Hyperbolic Equations11.4. Partial Functional Differential Equations with Arguments of GeneralType11.4.1. Parabolic Equations11.4.2. Hyperbolic Equations11.5. PDEs with Anisotropic Time Delay11.5.1. Parabolic Equations11.5.2. Hyperbolic EquationsReferences for Chapter 11