1 Algebraic and Transcendental Equations 1.1. Algebraic Equations 1.1.1. LinearandQuadraticEquations 1.1.2.

Cubic Equations 1.1.3. EquationsoftheFourthDegree 1.1.4. EquationsoftheFifthDegree 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 y x ''' x = f ( x, y ) 2.3.

2. Equations of the Form f ( x, y ) y x ''' x = g ( x, y, y x '' ) 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 ux + f ( x, y, u ) u y = 0 4.2.4. Equations of the Form ux + 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 '1 ur ) 5.1.5. Nonhomogeneous Heat Equation with Axial Symmetry u t = a ( u rr + r '1 u r ) + Φ( r, t ) 5.1.6. Heat Equation with Central Symmetry u t = a ( u rr + 2 r '1 ur ) 5.1.

7. Nonhomogeneous Heat Equation with Central Symmetry u t = a ( u rr + 2 r '1 u r ) + Φ( r, t ) 5.1.8. Heat Type Equation of the Form u t = u xx + (1 ' 2 β ) x '1 ux 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 ] x q ( x ) u + Φ( x, t ) 5.1.11. ' Liquid-Film Mass Transfer Equation (1 y 2) u x = au yy 5.1.12.

Equations of the Diffusion (Thermal) Boundary Layer n2 5.1.13. t 2 m xx Schro¨dinger Equation i n u = ' u + U ( x ) u 5.2. Hyperbolic Equations 5.2.1.

Wave Equation utt = a 2 uxx 5.2.2. Nonhomogeneous Wave Equation u tt = a 2 uxx + Φ( x, t ) 5.2.3. ' Klein-Gordon Equation u tt = a 2 uxx bu 5.2.

4. Nonhomogeneous Klein-Gordon Equation ' u tt = a 2 u xx bu + Φ( x, t ) 5.2.5. Wave Equation with Axial Symmetry u tt = a 2( u rr + r '1 u r ) + Φ( r, t ) 5.2.6. Wave Equation with Central Symmetry u tt = a 2( u rr + 2 r '1 u 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 = a 2 uxx + 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 + a 2 u xxxx = 0 5.6.2. Nonhomogeneous Equation of the Form u tt + a 2 u 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-.