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.