Equations of the Form ∂ 2 w ∂ 2 w
7.2.1. Equations of the Form ∂ 2 w ∂ 2 w
F (x, y)
∂x 2 ∂y 2
◮ Suppose w(x, y) is a solution of the equation in question. Then the function
w 1 = w(x, y) + C 1 xy + C 2 x+C 3 y+C 4 , where C 1 , ...,C 4 are arbitrary constants, is also a solution of the equation.
1. 2 = f (x)y 2 k .
∂x ∂y
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the function
1 k−2 w(x, C 1 y) + C 2 xy + C 3 x+C 4 y+C 5 ,
where C 1 , ...,C 5 are arbitrary constants, is also a solution of the equation.
2 ◦ . Generalized separable solutions:
Z x ( x − t)f (t)
w(x, y) = (C 1 x+C 2 ) y k+1 +
1 x+C 2 w(x, y) = (C x − t)f (t) ) y + dt + C 3 xy + C 4 x+C 5 y+C 6 ,
( k + 1)(k + 2) 0 ( C 1 t+C 2 ) where C 1 , ...,C 6 are arbitrary constants.
3 ◦ . Generalized separable solution:
k+2
w(x, y) = ϕ(x)y 2 + C 1 xy + C 2 x+C 3 y+C 4 ,
where C 1 , ...,C 4 are arbitrary constants and the function ϕ = ϕ(x) is determined by the ordinary differential equation
k(k + 2)ϕϕ ′′ xx =4 f (x).
2. 2 2 = f (x)g(y).
∂x ∂y
1 ◦ . Additive separable solution:
w(x, y) = C 1 ( x − t)f (t) dt + C 2 x+
( y − τ )g(τ ) dτ + C 3 y+C 4 ,
0 C 1 0 where C 1 , ...,C 4 are arbitrary constants.
2 ◦ . Multiplicative separable solution:
w(x, y) = ϕ(x)ψ(y),
where the functions ϕ = ϕ(x) and ψ = ψ(y) are determined by the ordinary differential equations ( C 1 is an arbitrary constant)
ϕϕ ′′ xx = C 1 f (x),
ψψ yy ′′
= −1 C
1 g(y).
3. ∂x 2 ∂y 2
= f (ax + by).
Solutions:
1 p ( z − t)
w(x, y) = ✢
where C 1 , ...,C 4 are arbitrary constants.
= f (x)y + g(x)y + h(x)y .
4. 2 k
k–1
∂y Generalized separable solution:
∂x 2 2
w(x, y) = ϕ(x)y k+1 + ψ(x)y + χ(x),
where the functions ϕ = ϕ(x), ψ = ψ(x), and χ = χ(x) are determined by the system of ordinary differential equations
k(k + 1)ϕϕ ′′ xx = f (x), k(k + 1)ϕψ ′′ xx = g(x), k(k + 1)ϕχ ′′ xx = h(x).
5. ∂x 2 ∂y 2
= f (x)e λy .
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the function
w 1 = C 1 w x, y − ln | C 1 | + C 2 xy + C 3 x+C 4 y+C 5 ,
λ where C 1 , ...,C 5 are arbitrary constants, is also a solution of the equation.
2 ◦ . Generalized separable solution:
1 Z x ( x − t)f (t)
w(x, y) = (C 1 x+C 2 ) e λy + 2 dt + C 3 xy + C 4 x+C 5 y+C 6 ,
λ x 0 C 1 t+C 2
where C 1 , ...,C 6 are arbitrary constants, and x 0 is any number such that the integrand does not
have a singularity at x=x 0 .
3 ◦ . Generalized separable solution:
w(x, y) = ϕ(x)e λy/2 + C 1 xy + C 2 x+C 3 y+C 4 ,
where C 1 , ...,C 4 are arbitrary constants, and the function ϕ = ϕ(x) is determined by the ordinary
differential equation 2 λ ϕϕ ′′ xx =4 f (x).
6. = f (x)e 2 λy + g(x)e λy .
∂x 2 ∂y 2
Generalized separable solution:
w(x, y) = ϕ(x)e λy + 2 ( x − t)
g(t)
dt + C 1 xy + C 2 x+C 3 y+C 4 ,
λ x 0 ϕ(t)
where C 1 , ...,C 4 are arbitrary constants and the function ϕ = ϕ(x) is determined by the ordinary differential equation 2 λ ϕϕ ′′
xx = f (x).
∂ 2 7.2.2. Monge–Amp `ere equation 2 w
–∂ w ∂ 2 2 w 2 =
F (x, y)
∂x∂y
∂x ∂y
Preliminary remarks.
The Monge–Amp`ere equation is encountered in differential geometry, gas dynamics, and meteorol- ogy.
1 ◦ . Suppose w(x, y) is a solution of the Monge–Amp`ere equation. Then the function
w 1 = w(x, y) + C 1 x+C 2 y+C 3 ,
where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.
2 ◦ . The transformation
2 x=a −2 ¯
1 x+b 1 y+c 1 , y=a ¯ 2 x+b 2 y+c 2 , w = kw + a ¯ 3 x+b 3 y+c 3 , F=k ¯ ( a 1 b 2 − a 2 b 1 ) F, where the a n , b n , c n , and k are arbitrary constants, takes the Monge–Amp`ere equation to an equation
of the same form.
3 ◦ . The transformation
x = x(1 + αx + βy) 4 ¯ ,¯ y = y(1 + αx + βy) , w = w(1 + αx + βy) ¯ , ¯ F = F (1 + αx + βy) , where α and β are arbitrary constants, takes the Monge–Amp`ere equation to an equation of the same
References : S. V. Khabirov (1990), N. H. Ibragimov (1994).
4 ◦ . In Lagrangian coordinates, the system of equations of the one-dimensional gas dynamics with plane waves is as follows:
where t is time, u the velocity, p the pressure, ξ the Lagrangian coordinate, and V the specific volume. The equation of state is assumed to have the form
V = V p, S(ξ) , where S = S(ξ) is a prescribed entropy profile. The Martin transformation
∂w
∂w
u(ξ, t) =
( x, y), t=
( x, y), x = ξ, y = p(ξ, t)
∂x
∂y
reduces the equations of the one-dimensional gas dynamics to the nonhomogeneous Monge–Amp `ere equation
F (x, y) = − ∂V ∂p p, S(ξ) .
References : M. N. Martin (1953), B. L. Rozhdestvenskii and N. N. Yanenko (1983).
Homogeneous Monge–Amp`ere equation .
1 ◦ . Suppose w(x, y) is a solution of the homogeneous Monge–Amp`ere equation. Then the functions w 1 = C 1 w(C 2 x+C 3 y+C 4 , C 5 x+C 6 y+C 7 )+ C 8 x+C 9 y+C 10 ,
w 2 = (1 + C 1 x+C 2 y)w
1+ C 1 x+C 2 y 1+ C 1 x+C 2 y
where C 1 , ...,C 10 are arbitrary constants, are also solutions of the equation.
where Φ 1 ( u, v) and Φ 2 ( u, z) are arbitrary functions of two arguments.
3 ◦ . General solution in parametric form:
w = tx + ϕ(t)y + ψ(t), x+ϕ ′ ( t)y + ψ ′ ( t) = 0,
where t is the parameter, and ϕ = ϕ(t) and ψ = ψ(t) are arbitrary functions.
4 ◦ . Solutions involving one arbitrary function:
w(x, y) = ϕ(C 1 x+C 2 y) + C 3 x+C 4 y+C 5 , y
w(x, y) = (C 1 x+C 2 y)ϕ
+ C 3 x+C 4 y+C 5 ,
1 2 3 C 4 x+C 5 y+C w(x, y) = (C 6 x+C y+C ) ϕ + C 7 x+C 8 y+C 9 ,
C 1 x+C 2 y+C 3
where C 1 , ...,C 9 are arbitrary constants and ϕ = ϕ(z) is an arbitrary function.
5 ◦ . Solutions involving arbitrary constants:
w(x, y) = C 1 y
C 2 2 + 2 xy + x + C 3 y+C 4 x+C 5 ,
w(x, y) =
w(x, y) = ✥ ( C 1 x+C 2 y+C 3 ) k + C 4 x+C 5 y+C 6 , ( C 1 x+C 2 y+C 3 ) k+1
w(x, y) = ✥
4 5 6 C x+C 8 y+C 9 ( , C x+C y+C )
y + b) 2 + C 5 x+C 6 y+C 7 , where the ✦✂✧
w(x, y) = ✥
x + a) 2 + C 2 ( x + a)(y + b) + C 3 (
a, b, and the C n are arbitrary constants.
References for equation 7.2.2.1: E. Goursat (1933), S. V. Khabirov (1990), N. H. Ibragimov (1994).
. First integrals for 2 A=a > 0:
where the Φ n ( u, v) are arbitrary functions of two arguments (n = 1, 2).
2 2 ◦ . General solution in parametric form for A=a > 0: β−λ
where β and λ are the parameters, ϕ = ϕ(β) and ψ = ψ(λ) are arbitrary functions.
w(x, y) = ★
x(C 1 x+C 2 y) + ϕ(C 1 x+C 2 y) + C 3 x+C 4 y,
w(x, y) = C 1 y + C 2 xy +
( C 2 − A)x + C 3 y+C 4 x+C 5 ,
w(x, y) =
w(x, y) = ★
( C 1 x−C 2 y + C 3 ) /2 + C 4 x+C 5 y+C 6 ,
where C 1 , ...,C 6 are arbitrary constants and ϕ = ϕ(z) is an arbitrary function. Another five solutions can be obtained:
(a) from the solution of equation 7.2.2.18 with α = 0 and f (u) = A, where β is an arbitrary constant; (b) from the solution of equation 7.2.2.20 with f (u) = A, where a, b, and c are arbitrary constants;
(c) from the solution of equation 7.2.2.21 with f (u) = A, where a, b, c, k, and s are arbitrary constants; (d) from the solution of equation 7.2.2.22 with α = 0 and f (u) = A, where β is an arbitrary constant; (e) from the solution of equation 7.2.2.24 with α = 0 and f (u) = A, where β is an arbitrary constant.
4 ◦ . The Legendre transformation
∂w
∂w
u = xξ + yη − w(x, y), ξ=
where u = u(ξ, η) is the new independent variable, and ξ and η are the new dependent variables, leads to an equation of the similar form
Reference : E. Goursat (1933).
= f (x).
∂x∂y
∂x 2 ∂y 2
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the functions
w −1
1 = C 1 w(x, C 2 x ★ C 1 y+C 3 )+ C 4 x+C 5 y+C 6 ,
where C 1 , ...,C 6 are arbitrary constants, are also solutions of the equation.
2 ◦ . Generalized separable solutions quadratic in y:
w(x, y) = C 1 y + C 2 xy +
( x − t)f (t) dt + C 3 y+C 4 x+C 5 ,
w(x, y) =
C 2 y + C 3 y+
( x − t)(t + C 1 ) f (t) dt + C 4 y+C 5 x+C 6 ,
x+C 1 4 C 2 2 C 2 0
where ✩✂✪ C 1 , ...,C 6 are arbitrary constants.
Reference : A. D. Polyanin and V. F. Zaitsev (2002).
3 ◦ . Generalized separable solutions for f (x) > 0:
Zp
w(x, y) = ★ y
f (x) dx + ϕ(x) + C 1 y,
where ✩✂✪ ϕ(x) is an arbitrary function.
References : M. N. Martin (1953), B. L. Rozhdestvenskii and N. N. Yanenko (1983).
D 2 x y(w x w yy − w y w xy + g y )− g−w x w y + D y y(w y w xx − w x w xy + g x )+( w x ) = 0,
where D x =
f (x) dx + ϕ(x) + ψ(y), and ϕ(x) and ψ(y) are arbitrary ∂x
Reference : S. V. Khabirov (1990).
5 ◦ . Let us consider some specific functions f = f (x). Solutions that can be obtained by the formulas of Items 1 ◦ and 2 ◦ are omitted.
5.1. Solutions with
f (x) = Ax k can be obtained:
(a) from the solution of equation 7.2.2.18 with f (u) = A and α = k/2, where β is an arbitrary constant; (b) from the solution of equation 7.2.2.24 with
f (u) = A and α = k/2, where β is an arbitrary constant. 5.2. Solutions for
f (x) = Ae λx :
w(x, y) = ✭ 2
A e λx/2 1 2 3 4 5 C 6
2 sin( C x+C y+C )+ C x+C y+C λ ,
w(x, y) =
e λx/2 sinh( C 1 x+C 2 y+C 3 )+ C 4 x+C 5 y+C 6 ,
w(x, y) =
y+C 6
e λx/2 cosh( C 1 x+C 2 y+C 3 )+ C 4 x+C 5 .
Another solution can be obtained from the solution of equation 7.2.2.22 with α = λ and f (u) = A, where β is an arbitrary constant.
= f (x)y.
∂x∂y
∂x 2 ∂y 2
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the functions
1 = C 1 w(x, C 1 y) + C 2 x+C 3 y+C 4 ,
where C 1 , ...,C 4 are arbitrary constants, are also solutions of the equation.
2 ◦ . Generalized separable solution quadratic in y:
w(x, y) = C 2
2 C 1 where C 1 , ...,C 5 are arbitrary constants.
3 ◦ . Generalized separable solution quadratic in y:
w(x, y) = ϕ(x)y 2 + ψ(x)y + χ(x),
where
1 f (x) dx ϕ(x) =
1 ϕ(x)
f (x) dx
, ψ(x) = C 3 ϕ(x) + C 4 +
C x+C
C ϕ(x)]
C 1 [ ϕ(x)]
x − t) t ( ( t)] dt + C 5 x+C 6 .
χ(x) =
2 a ϕ(t)
4 ◦ . Generalized separable solutions cubic in y:
1 3 w(x, y) = C 1 y −
( x − t)f (t) dt + C 2 x+C 3 y+C 4 ,
1 w(x, y) = 2 2 − ( x − t)(C t+C 2 ) f (t) dt + C 3 x+C 4 y+C 5 ,
( C 1 x+C 2 )
where C 1 , ...,C 5 are arbitrary constants.
5 ◦ . See solution of equation 7.2.2.7 in Item 3 ✫✂✬ ◦ with k = 1.
Reference : A. D. Polyanin and V. F. Zaitsev (2002).
= f (x)y 2 .
∂x∂y
∂x 2 ∂y 2
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the function
w −2 1 = ✯ C
1 w(x, C 1 y) + C 2 x+C 3 y+C 4 ,
where C 1 , ...,C 4 are arbitrary constants, is also a solution of the equation.
2 ◦ . Generalized separable solution quadratic in y:
w(x, y) = ϕ(x)y 3 + C
1 ϕ ( x) dx + C 2 y+ C 1 ( x − t)ϕ ( t) dt + C 3 x+C 4 ,
where the function ϕ = ϕ(x) is determined by the ordinary differential equation
xx = 2( ϕ x ′ ) − 2 f (x).
3 ◦ . Generalized separable solutions in the form of polynomials of degree 4 in y:
w(x, y) = C 1 y −
( x − t)f (t) dt + C 2 x+C 3 y+C 4 ,
12 C 1 a
w(x, y) = 3
( x − t)(C 1 t+C 2 )
where C 1 , ...,C 5 are arbitrary constants.
✰✂✱ 4 ◦ . See solution of equation 7.2.2.7 in Item 3 ◦ with k = 2.
Reference : A. D. Polyanin and V. F. Zaitsev (2002).
= f (x)y 2 + g(x)y + h(x).
∂x∂y
∂x 2 ∂y 2
Generalized separable solution quadratic in y:
w(x, y) = ϕ(x)y 2 + ψ(x)y + χ(x),
where the functions ϕ = ϕ(x), ψ = ψ(x), and χ = χ(x) are determined by the system of ordinary differential equations
xx = 2( ϕ x ) − 2 f (x),
xx ′′ =2 ϕ ′ x ψ ′ x − 2 g(x),
xx = 2 ( ψ ′ 2 x 1 ) − 2 h(x).
= f (x)y k .
∂x∂y
∂x 2 ∂y 2
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the functions
1 k−2 w(x, C 1 y) + C 2 x+C 3 y+C 4 ,
where C 1 , ...,C 4 are arbitrary constants, are also solutions of the equation.
2 ◦ . Generalized separable solutions:
C 1 y k+2
w(x, y) =
( x − t)f (t) dt + C 2 x+C 3 y+C 4 ,
( k + 1)(k + 2) C 1 a
y k+2
w(x, y) =
( x − t)(C 1 t+C 2 ) k+1 f (t) dt + C 3 x+C 4 y+C 5 ,
( C 1 x+C 2 ) k+1
( k + 1)(k + 2) a
where C 1 , ...,C 5 are arbitrary constants.
k+2
w(x, y) = ϕ(x)y 2 ,
where the function ϕ = ϕ(x) is determined by the ordinary differential equation
2 k(k + 2)ϕϕ 2 ′′
xx −( k + 2) ( ϕ ′ x ) +4 f (x) = 0.
4 ◦ . Let us consider the case where f is a power-law function of x, f (x) = Ax n , in more detail. Solutions:
C 1 x n+2
Ay k+2
w(x, y) =
+ C 2 y+C 3 x+C 4 ,
( n + 1)(n + 2) C 1 ( k + 1)(k + 2)
C 1 x n+2
Ay k+n+3
w(x, y) =
+ C 2 y+C 3 x+C 4 , ( n + 1)(n + 2)y
n+1 −
C 1 ( k + n + 2)(k + n + 3)
C 1 y k+2
Ax k+n+3
w(x, y) = ( 2 k + 1)(k + 2)x k+1 − +
C y+C x+C
( k + n + 2)(k + n + 3)
w(x, y) = (C − 1 x+C 2 ) k−1
y k+2 −
( x − t)t n ( C 1 t+C 2 ) k+1 dt + C 3 y+C 4 x,
( k + 1)(k + 2) a
w(x, y) = (C 1 y+C 2 ) n−1 x n+2 −
( y − t)t k ( C 1 t+C 2 ) n+1 dt + C 3 y+C 4 x,
( n + 1)(n + 2) a where C 1 , ...,C 4 are arbitrary constants.
There are also a multiplicative separable solution, see Item 3 ◦ with f (x) = Ax n , and a solution of the same type:
n+2
w(x, y) = ψ(y)x 2 ,
where the function ψ = ψ(y) is determined by the ordinary differential equation
yy −( n + 2) ( ψ y ) +4 Ay = 0.
n(n + 2)ψψ 2 ′′ ′ 2 k
The substitution − ψ=U n/2 brings it to the Emden–Fowler equation
yy ′′ = n 2 y U ( , n + 2)
n+1
whose exact solutions for various values of k and n can be found in the books by Polyanin and Zaitsev (1995, 2003).
Another exact solution for f (x) = Ax n can be obtained from the solution of equation 7.2.2.18 with f (u) = Au k ✲✂✳ and n = 2α + kβ, where α and β are arbitrary constants.
Reference : A. D. Polyanin and V. F. Zaitsev (2002).
= f (x)y 2 k+2 + g(x)y k .
∂x∂y
∂x 2 ∂y 2
Generalized separable solution:
k+2
g(t)
w(x, y) = ϕ(x)y
( x − t)
dt + C 1 x+C 2 y+C 3 ,
( k + 1)(k + 2) a ϕ(t)
where the function ϕ = ϕ(x) is determined by the ordinary differential equation
( 2 k + 1)(k + 2)ϕϕ ′′
xx −( k + 2) ( ϕ ′ 2 x ) + f (x) = 0.
Reference : A. D. Polyanin and V. F. Zaitsev (2002).
2 2 = f (x)e ∂x∂y . ∂x ∂y
λy
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the functions
w 1 = ✴ C 1 w x, y −
ln | C 1 | + C 2 x+C 3 y+C 4 ,
λ where C 1 , ...,C 4 are arbitrary constants, are also solutions of the equation.
2 ◦ . Generalized separable solutions:
w(x, y) = C 1 ( x − t)f (t) dt + C 2 x−
e λy + C 3 y+C 4 ,
w(x, y) = C 1 e − βx+λy − 2 ( x − t)e βt f (t) dt + C 2 x+C 3 y+C 4 ,
C 1 λ a where C 1 , ...,C 4 and β are arbitrary constants.
3 ◦ . Multiplicative separable solution:
w(x, y) = ϕ(x) exp 1
2 λy ,
where the function ϕ = ϕ(x) is determined by the ordinary differential equation
Reference : A. D. Polyanin and V. F. Zaitsev (2002).
= f (x)e 2 2 λy 2 + g(x)e λy .
∂x∂y
∂x ∂y
Generalized separable solution:
w(x, y) = ϕ(x)e λy
λ a ϕ(t)
where the function ϕ = ϕ(x) is determined by the ordinary differential equation
Reference : A. D. Polyanin and V. F. Zaitsev (2002).
= f (x)g(y) + A 2
∂x∂y
∂x 2 ∂y 2
Generalized separable solutions:
w(x, y) = C 1 ( x − t)f (t) dt −
( y − ξ)g(ξ) dξ ✴ Axy + C 2 x+C 3 y+C 4 ,
a C 1 b where C 1 , ...,C 4 are arbitrary constants; a and b are any numbers.
= f (ax + by).
w(x, y) = ✴
f (z) dz + ϕ(z) + C 1 x+C 2 y,
z = ax + by,
b where C 1 and C 2 are arbitrary constants and ϕ(z) is an arbitrary function.
2 ◦ . The transformation
w = U (x, z), z = ax + by
leads to an equation of the form 7.2.2.3:
= −2 + b
f (z).
Here, x and z play the role of y and x in 7.2.2.3, respectively.
=x k
f (ax + by).
The transformation w = U (x, z), z = ax + by
leads to an equation of the form 7.2.2.7:
2 2 + b x k f (z).
∂x∂z
∂x ∂z
Here, x and z play the role of y and x in 7.2.2.7, respectively.
=x 2 2 k+2 2 f (ax + by) + x k g(ax + by). ∂x∂y
∂x ∂y
The transformation w = U (x, z), z = ax + by
leads to an equation of the form 7.2.2.8:
2 2 + b x k+2 f (z) + b x k ∂x∂z g(z). ∂x ∂z Here, x and z play the role of y and x in 7.2.2.8, respectively.
f (ax + by).
The transformation w = U (x, z), z = ax + by
leads to an equation of the form 7.2.2.9:
= −2 + b e λx f (z).
∂x∂z
∂x 2 ∂z 2
Here, x and z play the role of y and x in 7.2.2.9, respectively.
=e 2 2 λx 2 f (ax + by) + e λx g(ax + by). ∂x∂y
∂x ∂y
The transformation w = U (x, z), z = ax + by
leads to an equation of the form 7.2.2.10:
g(z). Here, x and z play the role of y and x in 7.2.2.10, respectively.
This is a special case of equation 7.2.2.18 with α = −2 and β = −1.
1 ◦ . Suppose w(x, y) is a solution of the equation in question. Then the functions w 1 = ✷ w(C 1 x, C 1 y) + C 2 x+C 3 y+C 4 , where C 1 , ...,C 4 are arbitrary constants, are also solutions of the equation.
x where
∂x
∂y
C is an arbitrary constant.
References : M. N. Martin (1953), B. L. Rozhdestvenskii and N. N. Yanenko (1983).
where ϕ(z) is an arbitrary function.
4 ◦ . Conservation law:
∂x ∂x∂y
where D x =
Reference : S. V. Khabirov (1990).
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the functions
1 w(C 1 x, C 1 y) + C 2 x+C 3 y+C 4 ,
where C 1 , ...,C 4 are arbitrary constants, are also solutions of the equation.
2 ◦ . Self-similar solution:
w(x, y) = x α−β+1 u(z) z = x β y,
where the function u = u(z) is determined by the ordinary differential equation
2 [ 2 β(β + 1)zu ′ z +( α − β)(β − α − 1)u]u ′′ zz +( α + 1) ( u ′ z ) − f (z) = 0.
Reference : S. V. Khabirov (1990).
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the functions w 1 = w(x + 2bC 1 y + abC 1 , y + aC 1 )+ C 2 x+C 3 y+C 4 ,
where C 1 , ...,C 4 are arbitrary constants, are also solutions of the equation.
2 ◦ . Solutions:
Zp
w(x, y) = 2 F (z) + C
1 dz + C 2 x+C 3 y+C 4 ,
F (z) = 2 f (z) dz, z = ax − by ,
a b where C 1 , ...,C 4 are arbitrary constants.
Reference : A. D. Polyanin and V. F. Zaitsev (2002).
Solution for 2 b ≠4 ac:
2 w(x, y) = u(z) z = ax 2 + bxy + cy ,
where the function u = u(z) is determined by the ordinary differential equation
z u ′′ zz + (4 ac − b )( u ′ 2 z ) + f (z) = 0. Integrating yields
u(z) = ✻
dz + C 1 ,
F (z) = 2 f (z) dz + C 2 ,
b −4 ac where C 1 and C ✼✂✽ 2 are arbitrary constants.
Reference : A. D. Polyanin and V. F. Zaitsev (2002).
2 w(x, y) = u(z), 2 z = ax + bxy + cy + kx + sy,
where the function u = u(z) is determined by the ordinary differential equation
2 2 2 2 2 2 (4 ac − b ) z + as + ck − bks u ′ z u ′′ zz + (4 ac − b )( u ′ z ) + f (z) = 0. ′ The substitution 2
V (z) = (u z ) leads to a first-order linear equation.
Reference : A. D. Polyanin and V. F. Zaitsev (2002).
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the functions
1 w(x − 2 ln C 1 , C 1 y) + C 2 x+C 3 y+C 4 , where C 1 , ...,C 4 are arbitrary constants, are also solutions of the equation.
2 ◦ . Generalized self-similar solution:
2 α − β, where the function U = U (z) is determined by the ordinary differential equation
w(x, y) = e µx U (z), z=e βx
=e ky/x
f (x).
w(x, y) = exp
ϕ(x),
where the function ϕ = ϕ(x) is determined by the ordinary differential equation
2 x 2 ϕϕ
xx − x ( ϕ ′ 2 −2 x 4 ) +2 xϕϕ ′ x − ϕ +4 k x f (x) = 0.
w(x, y) = x α+2 u(z), z=x β e y/x ,
where the function u = u(z) is determined by the ordinary differential equation
2 z 2 βzu ′ +( α + 2)(α + 1)u u ′′
zz
+ z β − (α + 1) zu
z +( α + 2)(α + 1)u u z + f (z) = 0.
Reference : S. V. Khabirov (1990).
2 2 =y exp(2αy –1 )f (xy + βy ∂x∂y ). ∂x ∂y Solution:
w = y exp(αy −2 ) ϕ(z) + C
1 y+C 2 x+C 3 , z = xy + βy , where C 1 , C 2 , and C 3 are arbitrary constants, and the function ϕ = ϕ(z) is determined by the ordinary
differential equation
Reference : S. V. Khabirov (1990).
◮ For exact solutions of the nonhomogeneous Monge–Amp`ere equation for some specific
F = F (x, y) (without functional arbitrariness), see Khabirov (1990) and Ibragimov (1994). The Cauchy problem for the Monge–Amp`ere equation is discussed in Courant and Hilbert (1989).
–∂ w ∂ 2 7.2.3. Equations of the Form w
= F x, y, w, ∂w ∂x ∂y
, ∂w
∂x∂y
∂x 2
∂y 2
2 2 ∂x∂y = f (x)w + g(x). ∂x ∂y
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the function
w 1 = w(x, y + C 1 x+C 2 ), where C 1 and C 2 are arbitrary constants, is also a solution of the equation.
2 ◦ . Generalized separable solution quadratic in y:
w = ϕ(x)y 2 + ψ(x)y + χ(x), where ϕ(x), ψ(x), and χ(x) are determined by the system of ordinary differential equations
xx + f (x)ϕ − 4(ϕ x ′ ) = 0,
2 ϕψ xx ′′ + f (x)ψ − 4ϕ ′ x ψ ′ x = 0,
xx + f (x)χ + g(x) − (ψ x ′ ) = 0.
Note that the second equation is linear in ψ and has a particular solution ψ = ϕ (hence, its general solution can be expressed via the particular solution of the first equation).
= f (x)w 2
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the function
w 1 = C 1 w(x, y + C 2 x+C 3 ),
where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.
w(x, y) = e λy u(x),
where λ is an arbitrary constant and the function u = u(x) is determined by the ordinary differential equation
f (x)u = 0.
= f (x)y n w k .
∂x∂y
∂x 2 ∂y 2
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the function
w 1 = C n+2 w(x, C k−2 y),
where
C is an arbitrary constant, is also a solution of the equation.
2 ◦ . Multiplicative separable solution with n ≠ −2 and k ≠ 2:
n+2
w(x, y) = y 2− k U (x),
where the function U (x) is determined by the ordinary differential equation
2 2 ( 2 n + 2)(n + k)U U
xx −( n + 2) ( U x ′ ) +( k − 2) f (x)U k = 0.
= f (x)e w .
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the function
C is an arbitrary constant, is also a solution of the equation.
2 ◦ . Multiplicative separable solution with k ≠ 2 and λ ≠ 0:
λy
w(x, y) = exp
U (x),
2− k
where the function U (x) is determined by the ordinary differential equation
′ 2 2 UU −2 xx −( U x ) +( k − 2) λ f (x)U k = 0. ∂ 2 w 2 ∂ 2 w ∂ 2 w
= f (w).
∂x∂y
∂x 2 ∂y 2
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the function
w 1 = w(A 1 x+B 1 y+C 1 , A 2 x+B 2 y+C 2 ),
| A 2 B 1 − A 1 B 2 | = 1, where C 1 , C 2 , and any three of the four constants A 1 , A 2 , B 1 , and B 2 are arbitrary, is also a solution
of the equation.
2 ◦ . Functional separable solution:
2 w(x, y) = u(z), 2 z = ax + bxy + cy + kx + sy,
where
a, b, c, k, and s are arbitrary constants and the function u = u(z) is determined by the ordinary differential equation
2 2 2 2 2 2 (4 ac − b ) z + as + ck − bks u z ′ u ′′ zz + (4 ac − b )( u ′ z ) + f (u) = 0.
w(x, y) = exp
u(x),
2− k
where the function u = u(x) is determined by the ordinary differential equation
x 2 uu ′′
xx −( xu ′
x − u) + λ x f (x)u k = 0.
This equation is used in meteorology for describing wind fields in near-equatorial regions.
1 ◦ . Suppose w(x, y) is a solution of the equation in question. Then the function
where C 1 , ...,C 7 are arbitrary constants, is also a solution of the equation.
2 ◦ . Solutions:
w = ϕ(x),
1 √ w= 2
a x + C) y + ϕ(x),
where ϕ(x) is an arbitrary function and C is an arbitrary constant.
4 aC 2 ( x+C 1 ) tanh( C 2 y+C 3 ),
w= 2
4 aC 2 ( x+C 1 ) coth( C 2 y+C 3 ),
w= 2
4 aC 2 ( x+C 1 ) tan( C 2 y+C 3 ),
where ϕ(x) is an arbitrary function and C 1 , C 2 , C 3 , and λ are arbitrary constants. The first solution is a solution in additive separable form and the other four are multiplicative separable solutions. ❁✂❂
Reference : E. R. Rozendorn (1984).
4 ◦ . Generalized separable solution quadratic in y:
w = F (x)y 2 + G(x)y + H(x),
where
F (x) =
, G(x) = −
H(x) =
( x − t) t
− aG(t)
dt + C 5 x+C 6 ,
2 0 F (t)
and C 1 , ...,C 6 are arbitrary constants.
5 ◦ . Generalized separable solution:
a w=C 1 exp( C 2 x+C 2 3 y) − x + C 4 x+C 5 .
6 ◦ . There are exact solutions of the following forms:
w(x, y) = |x| k+2
U (z), z = y|x| − k ;
w(x, y) = e − kx V (ξ), ξ = ye kx ; w(x, y) = x 2 W (η),
η = y + k ln |x|;
where k is an arbitrary constant.
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the function
w −1
1 = C 1 w(x, C 1 y+C 2 x+C 3 )+ C 4 x+C 5 ,
where C 1 , ...,C 5 are arbitrary constants, is also a solution of the equation.
2 ◦ . Solutions:
w = ϕ(x),
w=
f (x) dx + C y + ϕ(x),
w=C 1 exp( C 2 x+C 3 y) −
( x − t)f (t) dt + C 4 x+C 5 ,
where ϕ(x) is an arbitrary function and C is an arbitrary constant. For C 2 = 0, the last solution is an additive separable solution.
3 ◦ . Generalized separable solution quadratic in y:
w = ϕ(x)y 2 + ψ(x)y + χ(x),
where
C 3 ϕ(x) =
ψ 2 ′ ( t)] − f (t)ψ(t)
and C 1 , ...,C 6 are arbitrary constants. ∂ 2 w 2
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the functions w 1 = −1 C 1 w(C 1 x+C 2 y+C 3 , ❃ y+C 4 )+ C 5 y+C 6 ,
where C 1 , ...,C 6 are arbitrary constants, are also solutions of the equation.
2 ◦ . Generalized separable solution linear in x:
w(x, y) = ϕ(y)x + ψ(y),
where ψ(y) is an arbitrary function and the function ϕ(y) is defined implicitly by
C is an arbitrary constant.
3 ◦ . Additive separable solution:
w(x, y) = C 2
1 y + C 2 y+C 3 + z(x),
where the function z(x) is determined by the autonomous ordinary differential equation 2C 1 z xx ′′ +
f (z ′ x ) = 0. Its general solution can be written out in parametric form as
dt
t dt
x = −2C 1
z = −2C 1
f (t)
f (t)
∂w
∂w
u = xξ + yη − w(x, y), ξ=
where u = u(ξ, η) is the new dependent variable and ξ and η are the new independent variables, leads to an equation of the form 7.2.2.3:
The Legendre transformation
∂w
∂w
u = xξ + yη − w(x, y), ξ=
where u = u(ξ, η) is the new dependent variable and ξ and η are the new independent variables, leads to the simpler equation
For exact solutions of this equation, see Subsection 7.2.2.
7.2.4. Equations of the Form
= f (x, y) ∂ w ∂ w
∂x 2 + ∂y g(x, y) 2
∂x∂y
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the function
w 1 = C 1 w(x, C 2 y+C 3 )+ C 4 x+C 5 y+C 6 , where C 1 , ...,C 6 are arbitrary constants, is also a solution of the equation.
2 ◦ . Degenerate solutions involving arbitrary functions:
w(x, y) = ϕ(x) + C 1 y+C 2 , w(x, y) = ϕ(y) + C 1 x+C 2 ,
where C 1 and C 2 are arbitrary constants and ϕ = ϕ(z) is an arbitrary function.
3 ◦ . Generalized separable solution quadratic in y:
[ ϕ ′ ( t)]
w(x, y) = ϕ(x)y +[ C 1 ϕ(x) + C 2 ] y+
( x − t) t
dt + C 3 x+C 4 ,
2 0 f (t)ϕ(t)
where the function ϕ = ϕ(x) is determined by the ordinary differential equation
f (x)ϕϕ ′′ xx − 2( ϕ ′ 2 x ) = 0.
4 ◦ . Generalized separable solution involving an arbitrary power of y:
w(x, y) = ϕ(x)y k + C 1 x+C 2 y+C 3
where the function ϕ = ϕ(x) is determined by the ordinary differential equation
( 2 k − 1)f (x)ϕϕ
xx ′′ − k(ϕ ′ x ) = 0.
5 ◦ . Generalized separable solution involving an exponential of y:
w(x, y) = ϕ(x)e λy + C 1 x+C 2 y+C 3 ,
where C 1 , C 2 , C 3 , and λ are arbitrary constants and the function ϕ = ϕ(x) is determined by the ordinary differential equation
f (x)ϕϕ 2
xx ′′ −( ϕ ′ x ) = 0.
2. = f (x)
+ g(x).
∂x∂y
∂x 2 ∂y 2
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the functions
w 1 = ❄ −1 C 1 w(x, C 1 y+C 2 )+ C 3 x+C 4 y+C 5 , where C 1 , ...,C 5 are arbitrary constants, are also solutions of the equation.
2 ◦ . Generalized separable solution linear in y:
Zp
w(x, y) = ❄ y
g(x) dx + ϕ(x) + C 1 y,
where ϕ(x) is an arbitrary function.
3 ◦ . Generalized separable solution quadratic in y:
w(x, y) = ϕ(x)y +[ C 1 ϕ(x) + C 1 2 ] y+ ( x − t) t
C [ ϕ ′ ( t)] − g(t)
dt + C 3 x+C 4 ,
2 0 f (t)ϕ(t)
where C 1 , ...,C 4 are arbitrary constants and the function ϕ = ϕ(x) is determined by the ordinary differential equation
f (x)ϕϕ ′′ xx − 2( ϕ ′ 2 x ) = 0, which has a particular solution ϕ=C 6 .
3. = f (x)
+ g(x)y.
∂x∂y
∂x 2 ∂y 2
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the function
1 = C 1 w(x, C 1 y) + C 2 x+C 3 y+C 4 ,
where C 1 , ...,C 4 are arbitrary constants, is also a solution of the equation.
2 ◦ . Generalized separable solution cubic in y:
g(t)
w(x, y) = C 1 y + C 2 y−
( x − t)
dt + C 3 x+C 4 ,
6 C 1 a f (t) where C 1 , ...,C 4 are arbitrary constants.
A more general solution is given by
g(t) dt
w(x, y) = ϕ(x)y + C 1 y−
( x − t)
+ C 2 x+C 3 ,
6 a f (t)ϕ(t)
where the function ϕ = ϕ(x) is determined by the ordinary differential equation
2 f (x)ϕϕ ′′ xx − 3( ϕ ′ 2 x ) = 0.
3 ◦ . For an exact solution quadratic in y, see equation 7.2.4.5 with g 2 = g 0 = 0.
4 ◦ . See the solution of equation 7.2.4.6 in Item 3 ◦ with k = 1.
4. = f (x)
+ g(x)y .
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the functions
w −2
1 = ❄ C 1 w(x, C 1 y) + C 2 x+C 3 y+C 4 ,
where C 1 , ...,C 4 are arbitrary constants, are also solutions of the equation.
1 4 2 w(x, y) = C g(t) y + C y− ( x − t) dt + C 3 x+C 4 ,
12 C 1 a f (t) where C 1 , ...,C 4 are arbitrary constants.
A more general solution is given by
g(t) dt
w(x, y) = ϕ(x)y + C 1 y−
( x − t)
+ C 2 x+C 3 ,
12 a f (t)ϕ(t)
where the function ϕ = ϕ(x) is determined by the ordinary differential equation
3 f (x)ϕϕ ′′ xx − 4( ϕ ′ 2 x ) = 0.
3 ◦ . For an exact solution quadratic in y, see equation 7.2.4.5 with g 1 = g 0 = 0.
4 ◦ . See the solution of equation 7.2.4.6 in Item 3 ◦ with k = 2.
5. = f (x)
+g 2 (x)y +g 1 (x)y + g 0 (x).
∂x∂y
∂x 2 ∂y 2
Generalized separable solution quadratic in y:
w(x, y) = ϕ(x)y 2 + ψ(x)y + χ(x),
where the functions ϕ = ϕ(x), ψ = ψ(x), and χ = χ(x) are determined by the system of ordinary differential equations
2 f (x)ϕϕ 1 ′′ = 2( ϕ ′
xx
x ) − 2 g 2 ( x),
f (x)ϕψ ′′ xx =2 ϕ ′ x ψ x ′ 1 − 2 g 1 ( x),
f (x)ϕχ ′′
xx = 2 ( ψ ′ x ) − 2 g 0 ( x).
6. = f (x)
+ g(x)y k .
∂x∂y
∂x 2 ∂y 2
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the functions
1 k−2 w(x, C 1 y) + C 2 x+C 3 y+C 4 ,
where C 1 , ...,C 4 are arbitrary constants, are also solutions of the equation.
2 ◦ . Additive separable solution:
1 k+2
g(t)
w(x, y) =
( k + 1)(k + 2)
C 1 a f (t)
where C 1 , ...,C 4 are arbitrary constants.
3 ◦ . Multiplicative separable solution:
k+2
w(x, y) = ϕ(x)y 2 ,
where the function ϕ = ϕ(x) is determined by the ordinary differential equation
2 k(k + 2)f (x)ϕϕ 2 ′′
xx −( k + 2) ( ϕ x ′ ) +4 g(x) = 0.
4 ◦ . Generalized separable solution:
w(x, y) = ψ(x)y k+2 −
( k + 1)(k + 2) a f (t)ψ(t)
where the function ψ = ψ(x) is determined by the ordinary differential equation
( 2 k + 1)f (x)ψψ ′′
xx −( k + 2)(ψ x ′ ) = 0.
2 2 + g(x)y 2 k+2 + h(x)y k ∂x∂y . ∂x ∂y Generalized separable solution:
7. = f (x)
w(x, y) = ϕ(x)y k+2
h(t)
( x − t)
dt + C 1 x+C 2 y+C 3 , ( k + 1)(k + 2) a f (t)ϕ(t)
where the function ϕ = ϕ(x) is determined by the ordinary differential equation
x ) + g(x) = 0. ∂ 2 2 w
( 2 k + 1)(k + 2)f (x)ϕϕ ′′ −( k + 2) ( ϕ ′ 2
xx
8. = f (x)
+ g(x)e λy .
∂x∂y
∂x 2 ∂y 2
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the functions
w 1 = C 1 w x, y − ln | C 1 | + C 2 x+C 3 y+C 4 ,
where C 1 , ...,C 4 are arbitrary constants, are also solutions of the equation.
2 ◦ . Additive separable solution:
g(t) w(x, y) = C
where C 1 , ...,C 4 are arbitrary constants.
3 ◦ . Multiplicative separable solution:
w(x, y) = ϕ(x) exp 1
2 λy ,
where the function ϕ = ϕ(x) is determined by the ordinary differential equation
2 f (x)ϕϕ −2 ′′
xx −( ϕ x ′ ) +4 λ g(x) = 0.
4 ◦ . Generalized separable solution:
w(x, y) = ψ(x)e λy
a f (t)ψ(t)
where the function ψ = ψ(x) is determined by the ordinary differential equation
f (x)ψψ 2
xx ′′ −( ψ x ′ ) = 0.
9. 2 = f (x) λy
+ g(x)e
+ h(x)e .
Generalized separable solution:
w(x, y) = ϕ(x)e λy
1 h(t)
− 2 ( x − t)
dt + C 1 x+C 2 y+C 3 ,
λ a f (t)ϕ(t) where the function ϕ = ϕ(x) is determined by the ordinary differential equation
f (x)ϕϕ ′′ xx −( ϕ ′ 2 x −2 ) + λ g(x) = 0.
2 2 +f 2 (x)g 2 ∂x∂y (y). ∂x ∂y
10. =f 1 (x)g 1 (y)
1 ◦ . Suppose w(x, y) is a solution of this equation. Then the function
w 1 = w(x, y) + C 1 x+C 2 y+C 3 ,
where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.
2 ◦ . Additive separable solution for f 1 g 1 ≠ 0:
f 2 ( t)
g 2 ( ξ)
w(x, y) = C 1 ( x − t)
where C 1 , ...,C 4 are arbitrary constants.
3 ◦ . Degenerate solutions for f 2 g 2 = 0: w(x, y) = ϕ(x) + C 1 y+C 2 ,
w(x, y) = ϕ(y) + C 1 x+C 2 ,
where C 1 and C 2 are arbitrary constants and ϕ = ϕ(z) is an arbitrary function.
4 ◦ . Generalized separable solution for f 2 g 2 = 0: w(x, y) = ϕ(x)ψ(y) + C 1 x+C 2 y+C 3 ,
where the functions ϕ = ϕ(x) and ψ = ψ(y) are determined by the ordinary differential equations
f 2 1 ( x)ϕϕ ′′
xx − C 4 ( ϕ ′ x ) = 0,
4 g 1 ( y)ψψ ′′ yy −( ψ y ′ ) = 0.
11. = f (ax + by)
+ g(ax + by).
2 w(x, y) = ϕ(z) + C 2
1 x + C 2 xy + C 3 y + C 4 x+C 5 y, z = ax + by, where C 1 , ...,C 5 are arbitrary constants and the function ϕ(z) is determined by the ordinary
differential equation
2 2 2 ( 2 abϕ zz + C ) = f (z)(a ϕ ′′ zz +2 C 1 )( b ϕ ′′ zz +2 C 3 )+ g(z), which is easy to integrate; to this end, the equation should first be solved for ϕ zz ′′ .