Equations with Two Independent Variables, Nonlinear in Two or More Highest Derivatives

8.2.4. Equations with Two Independent Variables, Nonlinear in Two or More Highest Derivatives

f 2 =g 1 (x)g (y).

Generalized separable solution: w(x, y) = ϕ(x) + ψ(y) + C 1 xy + C 2 x+C 3 y+C 4 , where C 1 , ...,C 4 are arbitrary constants, and the functions ϕ = ϕ(x) and ψ = ψ(y) are determined

by the ordinary differential equations ( a is any)

f 1 ( ϕ ′′ xx )= ag 1 ( x),

af 2 ( ψ ′′ yy )= g 2 ( y).

The substitution u= ∂w ∂x leads to the first-order partial differential equation

For details about integration methods and exact solutions for such equations (with various

F ), see Kamke (1965) and Polyanin, Zaitsev, and Moussiaux (2002).

1 ◦ . Solution quadratic in both variables:

1 w(x, y) = 2

2 C 1 x C 2 xy + 2 1 F (C , C 2 ) y + C 3 x+C 4 y+C 5 ,

where C 1 , ...,C 5 are arbitrary constants.

2 ◦ . We differentiate the equation with respect to x, introduce the new variable

and then apply the Legendre transformation (for details, see Subsection S.2.3)

to obtain the second-order linear equation

where the subscripts

X and Y denote the corresponding partial derivatives.

Special case. Let

Solution: w = ϕ(z) + 1 ( A 2 A 3 − A 1 A 4 ) x 3 1 2 1 2 6 1 + 2 aA 1 A 3 x y+ 2 aA 2 A 3 xy + 6 ( a 2 A 1 A 3 + aA 2 A 4 ) y 3

+ 1 2 B 1 x 2 + B 2 xy + 1 2 B 3 y 2 + B 4 x+B 5 y+B 6 , z=A 1 x+A 2 y, where the A n and B m are arbitrary constants and the function ϕ(z) is determined by the ordinary differential equation

∂x∂y ∂y 2

1 ◦ . Solution quadratic in both variables:

2 w(x, y) = A 2

11 x + A 12 xy + A 22 y + B 1 x+B 2 y + C,

where A 11 , A 12 , A 22 , B 1 , B 2 , and

C are arbitrary constants constrained by F (2A 11 , A 12 ,2 A 22 ) = 0.

2 ◦ . Solving the equation for w yy (or w xx ), one arrives at an equation of the form 8.2.4.2.

∂x ∂x ∂x∂y

∂y ∂x∂y ∂y 2

Additive separable solution:

w(x, y) = ϕ(x) + ψ(y).

Here, ϕ(x) and ψ(y) are determined by the ordinary differential equations

C is an arbitrary constant.

Multiplicative separable solution:

w(x, y) = ϕ(x)ψ(y).

Here, the functions ϕ(x) and ψ(y) are determined by the ordinary differential equations

C is an arbitrary constant.

Multiplicative separable solution:

w(x, y) = ϕ(x)ψ(y).

Here, the functions ϕ(x) and ψ(y) are determined by the ordinary differential equations

1 x, ϕ ′ x /ϕ, ϕ ′′ xx /ϕ ψ k F 2 y, ψ ′ y /ψ, ψ ′′ yy /ψ

where

C is an arbitrary constant.

w = w(ξ), ξ = ax + by,

where the function w(ξ) is determined by the ordinary differential equation

The substitution u(x, y) = w(x, y) + kx + sy leads to an equation of the form 8.2.4.8:

∂u

∂u

F ax + by, u,

− k,

− s,

2 ∂y , ∂y 2 ,

∂x

∂x

∂x∂y

∂x ∂y ∂x 2 ∂x∂y ∂y 2

Traveling-wave solutions:

w = w(z),

z=a 2 x + (k − a 1 ) y,

where k is a root of the quadratic equation

k 2 −( a

1 + b 2 ) k+a 1 b 2 − a 2 b 1 = 0,

and the function w(z) is determined by the ordinary differential equation kzw ′ z =

, , . ∂x ∂y ∂x 2 ∂x∂y ∂y 2

Exact solutions are sought in the traveling-wave form

w = w(z),

z = Ax + By + C,

where the constants

A, B, and C are determined by solving the algebraic system

c 1 A k + c 2 B k = C. (3) Equations (1) and (2) are first solved for

A and B, and then C is evaluated from (3). The desired function w(z) is determined by the ordinary differential equation

∂x 2 ∂x∂y

, , , . ∂x ∂y ∂x 2 ∂x∂y ∂y 2

Traveling-wave solutions:

w = w(z),

z = Ax + By,

where the constants

A and B are determined by solving the algebraic system of equation

1 A + a 2 AB + a 3 B = A,

b 1 A 2 + b 2 AB + b 3 B 2 = B,

and the desired function w(z) is determined by the ordinary differential equation