Equations of the Form ∂w 2 = f (x, t) ∂ w +g x, t, w, ∂w

1.6.8. Equations of the Form ∂w 2 = f (x, t) ∂ w +g x, t, w, ∂w

∂t

∂x 2 ∂x

f (w).

The substitution z= √ 2 leads to an equation of the form 1.6.1.1:

ax + b

∂w

+ f (w).

f (t) ∂ n 2. ∂w = x + g(t)w ln w. ∂t

x n ∂x

∂x

This equation can be rewritten in the form

+ g(t)w ln w.

∂t

∂x 2 x ∂x

Functional separable solution:

w(x, t) = exp 2 ϕ(t)x + ψ(t) ,

where the functions ϕ(t) and ψ(t) are determined by the system of first-order ordinary differential equations with variable coefficients

ϕ ′ t =4 fϕ 2 + gϕ, ψ t ′ = 2( n + 1)f ϕ + gψ;

the arguments of f and g are omitted. Successively integrating, one obtains

ϕ(t) = e G A−4

ψ(t) = Be G + 2(

n + 1)e − G f ϕe G dt,

where

A and B are arbitrary constants.

+ ➳ ( t)w ln w + x 2 p(t) + q(t) w. ∂t

3. = f (t)

2 + xg(t) +

∂x

∂x

Functional separable solution:

w(x, t) = exp 2 ϕ(t)x + ψ(t) ,

where the functions ϕ(t) and ψ(t) are determined by the system of first-order ordinary differential equations with variable coefficients

2 t =4 fϕ + (2 g + s)ϕ + p,

(2) For p ≡ 0, equation (1) is a Bernoulli equation and, hence, can be easily integrated. In the

ψ ′ t =s ψ + 2(f + h)ϕ + q.

general case, (1) is a Riccati equation for ϕ = ϕ(t), so it can be reduced to a second-order linear equation. The books by Kamke (1977) and Polyanin and Zaitsev (2003) present a considerable number of solutions to this equation for various f , g, s, and p. Having solved equation (1), one can find ψ = ψ(t) from the linear equation (2).

+ g(t)w ln w + h(t)w.

∂t ∂x

∂x

Functional separable solution:

w(x, t) = exp − ϕ(t)e λx + ψ(t) ,

where the functions ϕ(t) and ψ(t) are determined by the ordinary differential equations

ϕ ′ = λ 2 f (t)ϕ t 2 + g(t)ϕ, ψ ′ t = g(t)ψ + h(t).

Integrating yields

ϕ(t) = G(t) A−λ 2 f (t)G(t) dt , G(t) = exp

g(t) dt ,

Z h(t)

ψ(t) = BG(t) + G(t)

A and B are arbitrary constants. ∂w

This equation can be rewritten in the form

1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function

w 1 = exp( C 1 e at ) w(x, t + C 2 ), where C 1 and C 2 are arbitrary constants, is also a solution of the equation.

2 ◦ . Multiplicative separable solution:

w(x, t) = exp(Ce at ) ϕ(x),

where

C is an arbitrary constant, and the function ϕ(t) is determined by the ordinary differential equation

( fϕ ′ x ) ′ x + aϕ ln ϕ = 0.

f (x)

+ aw ln w + g(x) + h(t) w.

∂t ∂x

∂x

1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function

w 1 = exp( Ce at ) w(x, t),

where

C is an arbitrary constant, is also a solution of the equation.

2 ◦ . Multiplicative separable solution:

w(x, t) = exp Ce at +

e − at e at h(t) dt ϕ(x),

where the function ϕ(x) is determined by the ordinary differential equation

( fϕ ′ x ) ′ x + aϕ ln ϕ + g(x)ϕ = 0.

2 + g(x)

+ aw ln w + h(x) + ➵ ( t) w.

∂t ∂x

∂x

1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function

w 1 = exp( Ce at ) w(x, t),

where

C is an arbitrary constant, is also a solution of the equation.

2 ◦ . Multiplicative separable solution:

w(x, t) = exp Ce at +

e − at e at s( t) dt ϕ(x),

where the function ϕ(x) is determined by the ordinary differential equation

f (x)ϕ ′′ xx + g(x)ϕ ′ x + aϕ ln ϕ + h(x)ϕ = 0. ∂w 2 ∂ 2 w ∂w

∂w

8. = f (x)

2 + g(x)

+ h(x)

+ aw + p(x) + q(t).

1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function

w 1 = w(x, t) + Ce at ,

where

C is an arbitrary constant, is also a solution of the equation.

2 ◦ . Additive separable solution:

w(x, t) = ϕ(x) + Ce at +

e − at e at q(t) dt,

where the function ϕ(x) is determined by the ordinary differential equation

f (x)ϕ ′′ xx + g(x)(ϕ x ′ ) 2 + h(x)ϕ ′ x + aϕ + p(x) = 0.

+ aw + p(x) + q(t). ∂t

9. = f (x)

+ g(x)

+ h(x)

∂x 2 ∂x

∂x

1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function

w 1 = w(x, t) + Ce at ,

where

C is an arbitrary constant, is also a solution of the equation.

2 ◦ . Additive separable solution:

w(x, t) = ϕ(x) + Ce at +

e − at e at q(t) dt,

where the function ϕ(x) is determined by the ordinary differential equation

f (x)ϕ ′′ xx + g(x)(ϕ ′ x ) k + h(x)ϕ ′ x + aϕ + p(x) = 0.

10. = f (x)

+ g x,

+ aw + h(t).

1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function

w 1 = w(x, t) + Ce at ,

where

C is an arbitrary constant, is also a solution of the equation.

2 ◦ . Additive separable solution:

w(x, t) = ϕ(x) + Ce at + e at

e − at h(t) dt,

where the function ϕ(x) is determined by the ordinary differential equation

f (x)ϕ ′′ xx + g(x, ϕ ′ x )+ aϕ = 0.