General Form Equations Involving the First
11.2. General Form Equations Involving the First
Derivative in t
11.2.1. Equations of the Form ∂w n = F w, ∂w , ...,∂ w ∂t n ∂x ∂x
Preliminary remarks. Consider the equation
1 ◦ . Suppose w(x, t) is a solution of equation (1). Then the function w(x + C 1 , t+C 2 ), where C 1 and C 2 are arbitrary constants, is also a solution of the equation.
2 ◦ . In the general case, equation (1) admits a traveling-wave solution
(2) where k and λ are arbitrary constants, and the function w(ξ) is determined by the ordinary differential
w = w(ξ),
ξ , ...,k n w n) ξ − λw ′ ξ = 0.
Special cases of equation (1) that admit, apart from traveling-wave solutions (2), also other types of solution are presented in this subsection.
1. =F ∂t
∂x n
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
where C 1 , C 2 , C 3 , and the A k are arbitrary constants, is also a solution of the equation.
2 ◦ . Additive separable solution:
···+C 1 x+C 0 , where
w(x, t) = F (A)t +
x n + C n−1 x n−1 +
n!
0 , A, C C 1 , ...,C n−1 are arbitrary constants.
3 ◦ . Solution linear in t:
w(x, t) = t
where the A k and B k are arbitrary constants and Φ( u) is the inverse of the function F (u).
4 ◦ . Solution:
1 X n−1
w(x, t) = A 1 t+
B x m m + U (z),
where A 1 , A 2 , the B m , k, and λ are arbitrary constants, and the function U = U (z) is determined by the autonomous ordinary differential equation
A 1 + λU z ′ = FA 2 + k n U ( n) z .
5 ◦ . Self-similar solution:
w(x, t) = t Θ(ζ), −1 ζ = xt /n ,
where the function Θ( ζ) is determined by the ordinary differential equation
w(x, t) = At + B + ϕ(ξ),
ξ = kx + λt,
where
A, B, k, and λ are arbitrary constants, and the function ϕ(ξ) is determined by the autonomous ordinary differential equation
F kϕ 2 ′
, ( k ϕ ′′ , ...,k n ϕ n) − λϕ ′
This is a special case of equation 11.2.2.1 with g(t) = a and F t = 0.
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
w 1 = w(x + aC 1 t+C 2 , t+C 3 )+ C 1 ,
where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.
2 ◦ . Degenerate solution:
x+C 1 1
w(x, t) = −
τF −
, 0, ...,0 dτ ,
τ=t+C 2 .
w(x, t) = U (ζ) + 2C 2 1 t, ζ = x + aC 1 t + C 2 t,
where C 1 and C 2 are arbitrary constants and the function U (ζ) is determined by the autonomous ordinary differential equation
FU ( n)
ζ ′ , U ζζ ′′ , ...,U ζ
+ aU U ζ ′ = C 2 U ζ ′ +2 C 1 .
In the special case C 1 = 0, we have a traveling-wave solution.
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
w 1 = w(x + aC 1 e bt + C 2 , t+C 3 )+ C 1 be bt ,
where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.
2 ◦ . There is a degenerate solution linear in x:
w(x, t) = ϕ(t)x + ψ(t).
3 ◦ . Traveling-wave solution:
w = w(ξ), ξ = x + λt,
where λ is an arbitrary constant and the function w(ξ) is determined by the autonomous ordinary differential equation
Fw ξ ′
, ( w ′′
ξξ , ...,w n) ξ + aww ′ ξ − λw ′ ξ + bw = 0.
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
w −1
1 = C 1 w(x + C 2 , C 1 t+C 3 ),
where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.
2 ◦ . Solution:
w(x, t) = tϕ(ξ), ξ = kx + λ ln |t|,
where k and λ are arbitrary constants, and the function ϕ(ξ) is determined by the autonomous ordinary differential equation
( n)
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
w 1 = C 1 w(x + C 2 , t+C 3 ),
where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.
2 ◦ . Multiplicative separable solution:
w(x, t) = Ce λt ϕ(x),
where
C and λ are arbitrary constants, and the function ϕ(x) is determined by the autonomous ordinary differential equation
F x ϕ ′′
n)
, xx , ..., x
This equation has particular solutions of the form ϕ(x) = e αx , where α is a root of the algebraic (or transcendental) equation
F α, α 2 , ...,α n − λ = 0.
3 ◦ . Solution:
w(x, t) = Ce λt ψ(ξ), ξ = kx + βt
where
C, k, λ, and β are arbitrary constants, and the function ψ(ξ) is determined by the autonomous ordinary differential equation
This equation has particular solutions of the form ψ(ξ) = e µξ . ∂w
For β = 0, see equation 11.2.1.6, and for β = 1, see 11.2.1.7.
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
w 1 = C 1 w(x + C 2 , β−1 C 1 t+C 3 ),
where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.
2 ◦ . Multiplicative separable solution:
w(x, t) = (1 − β)At + B 1− β ϕ(x),
where
A and B are arbitrary constants, and the function ϕ(x) is determined by the autonomous ordinary differential equation
ϕ F , xx , ..., x
= A.
3 ◦ . Solution:
w(z, t) = (t + C) 1− β Θ( z), z = kx + λ ln(t + C), where
C, k, and λ are arbitrary constants, and the function Θ(z) is determined by the autonomous ordinary differential equation
Θ , ...,k n Θ z Θ
Θ ( n)
zz
= λΘ z +
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
w 1 = w(x + C 1 , C 2 t+C 3 )+
ln C 2 ,
where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.
2 ◦ . Additive separable solution:
1 w(x, t) = − ln( Aβt + B) + ϕ(x),
where
A and B are arbitrary constants, and the function ϕ(x) is determined by the autonomous ordinary differential equation
e ( βϕ Fϕ ′
x , ϕ ′′
xx , ...,ϕ x +
1 w(x, t) = − ln( t + C) + Θ(ξ), ξ = kx + λ ln(t + C),
where
C, k, and λ are arbitrary constants, and the function Θ(ξ) is determined by the autonomous ordinary differential equation
e βΘ
F kΘ ′ ξ , k Θ ′′ ξξ , ...,k n Θ
This is a special case of equation 11.2.1.2.
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
w −1 1 = C
1 w(x + C 2 , C 1 t+C 3 )+ C 4 ,
where C 1 , ...,C 4 are arbitrary constants, is also a solution of the equation.
2 ◦ . Solution:
w(x, t) = At + B + ϕ(ξ),
ξ = kx + λt,
where
A, B, k, and λ are arbitrary constants, and the function ϕ(ξ) is determined by the autonomous ordinary differential equation
( F kϕ n) ′′
ξξ /ϕ , ...,k n−1 ξ ′ ϕ ξ /ϕ ′ ξ = λϕ ′ ξ + A.
3 ◦ . Solution: w(x, t) = (t + C 1 )Θ( z) + C 2 , z = kx + λ ln |t + C 1 |,
where C 1 , C 2 , k, and λ are arbitrary constants, and the function Θ(z) is determined by the autonomous ordinary differential equation
F kΘ ′′ zz /Θ ′ z , ...,k n−1 Θ ( n) z /Θ z ′ = λΘ ′ z + Θ.
This is a special case of equation 11.2.1.2.
w 1 = C 1 w(x + C 2 , t+C 3 )+ C 4 , where C 1 , ...,C 4 are arbitrary constants, is also a solution of the equation.
2 ◦ . Solution:
w(x, t) = At + B + ϕ(z),
z = kx + λt,
where
A, B, k, and λ are arbitrary constants, and the function ϕ(z) is determined by the autonomous ordinary differential equation
kϕ ′ z
F kϕ ′′ zz /ϕ ′ z , ...,k n−1 ϕ ( n) z /ϕ ′ z = λϕ ′ z + A.
3 ◦ . Solution:
w(x, t) = Ae βt Θ( ξ) + B,
ξ = kx + λt,
where
A, B, k, β, and λ are arbitrary constants, and the function Θ(ξ) is determined by the autonomous ordinary differential equation
F kΘ ′′ ξξ /Θ ′ ξ , ...,k n−1 Θ ( n) ξ /Θ ′ ξ = λΘ ′ ξ + βΘ. ∂w
This is a special case of equation 11.2.1.2. For β = 0 and β = 1 see equations 11.2.1.10 and 11.2.1.11.
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
w 1 = C 1 w(x + C 2 , C β−1 1 t+C 3 )+ C 4 , where C 1 , ...,C 4 are arbitrary constants, is also a solution of the equation.
2 ◦ . Generalized separable solution:
w(x, t) = A(1 − β)t + B 1− β ϕ(x) + C,
where
A, B, and C are arbitrary constants, and the function ϕ(x) is determined by the autonomous ordinary differential equation
Fϕ ′′ xx /ϕ ′ x , ...,ϕ n) x /ϕ ′ x = Aϕ.
3 ◦ . Solution:
w(x, t) = (t + A) 1− β Θ( z) + B, z = kx + λ ln(t + A), where
A, B, k, and λ are arbitrary constants, and the function Θ(z) is determined by the autonomous ordinary differential equation
F kΘ ′′
zz /Θ z ′ , ...,k n−1 Θ ( n) z /Θ ′ z = λΘ z ′ +
11.2.2. Equations of the Form ∂w n = F t, w, ∂w , ...,∂ w ∂t n ∂x ∂x
+ g(t)w.
Suppose w(x, t) is a solution of the equation in question. Then the function
w 1 = w(x, t) + C exp
g(t) dt ,
where
C is an arbitrary constant, is also a solution of the equation.
+ g(t).
The transformation
w = u(z, t) +
g(τ ) dτ ,
z=x+a
( t − τ )g(τ ) dτ ,
where t 0 is any, leads to a simpler equation of the form 11.2.1.4:
2 ∂x ,..., ∂x ∂x n + f (t)w
+ g(t)w.
∂t
∂x
Suppose w(x, t) is a solution of the equation in question. Then the function w 1 = wx+C 1 ψ(t) + C 2 , t + C 1 ϕ(t), where
f (t)ϕ(t) dt, C 1 and C 2 are arbitrary constants, is also a solution of the equation.
ϕ(t) = exp
g(t) dt , ψ(t) =
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
w 1 = C 1 w(x + C 2 , t),
where C 1 and C 2 are arbitrary constants, is also a solution of the equation.
2 ◦ . Multiplicative separable solution:
F (t, λ, . . . , λ n ) dt , where
w(x, t) = A exp λx +
A and λ are arbitrary constants. ∂w
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
w 1 = C 1 w(x + C 2 , t),
where C 1 and C 2 are arbitrary constants, is also a solution of the equation.
2 ◦ . Multiplicative separable solutions:
2 w(x, t) = A exp 2 λx + F t, λ , ...,λ n dt ,
2 w(x, t) = 2 A cosh(λx) + B sinh(λx) exp F t, λ , ...,λ n dt ,
2 w(x, t) = 2 A cos(λx) + B sin(λx) exp F t, −λ , . . . , (−1) n λ n dt , where
A, B, and λ are arbitrary constants.
2 ∂t ,..., w ∂x w ∂x w ∂x n + g(t)w.
6. = f (t)w Φ
The transformation
G(t) = exp g(t) dt , leads to a simpler equation of the form 11.2.1.8:
w(x, t) = G(t)u(x, τ ), τ=
f (t)G β−1 ( t) dt,
which has, for instance, a traveling-wave solution u = u(ax + bτ ) and a multiplicative solution of the form u = ϕ(x)ψ(τ ).
∂w
∂w ∂ 2 w
7. = f (t)e Φ
βw
+ g(t).
The transformation
Z w(x, t) = u(x, τ ) + G(t), τ=
G(t) = g(t) dt, leads to a simpler equation of the form 11.2.1.9:
f (t) exp βG(t) dt,
which has, for instance, a traveling-wave solution u = u(ax + bτ ) and an additive separable solution of the form u = ϕ(x) + ψ(τ ).
+ g(t)
The transformation
f (t) dt, leads to the simpler equation
w = u(z, τ ),
z=x+
g(t) dt,
∂u 2 ∂u ∂ u
2 , ∂τ ..., ∂z ∂z ∂z n ,
=Φ u,
which has a traveling-wave solution u = u(kz + λτ ). ∂w
w ∂x w ∂x n The equation has a multiplicative solution of the form
w(x, t) = Ae λx Θ( t),
where
A and λ are arbitrary constants. ∂w
10. =w Φ 0 t,
k Φ k w t, ∂x 2 ,..., .
w ∂x 2 w ∂x 2 n The equation has multiplicative solutions of the following forms:
w(x, t) = Ae λx Θ 1 ( t),
w(x, t) =
A cosh(λx) + B sinh(λx) Θ 1 ( t),
w(x, t) =
A cos(λx) + B sin(λx) Θ 2 ( t),
where
A, B, and λ are arbitrary constants.
∂t n ∂x ∂x
∂w
2 ∂t ,..., ∂x ∂x n .
1. =F x,
Generalized separable solution linear in t:
w(x, t) = Axt + Bt + C + ϕ(x),
where
A, B, and C are arbitrary constants, and the function ϕ(x) is determined by the ordinary differential equation
F x, ϕ ( ′′
xx , ...,ϕ n) x = Ax + B.
Additive separable solution:
w(x, t) = At + B + ϕ(x),
where
A and B are arbitrary constants, and the function ϕ(x) is determined by the ordinary differential equation
F x, ϕ ′ x , ϕ ′′ xx , ...,ϕ ( n) x = A.
∂w ∂w
n–1 ∂ n 3. w =F ,x ,...,x .
∂t ∂x
∂x 2 ∂x n
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
w −1
1 = C 1 w(C 1 x, C 1 t+C 2 )+ C 3 ,
where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.
2 ◦ . Additive separable solution:
w(x, t) = At + B + ϕ(x),
where
A and B are arbitrary constants, and the function ϕ(x) is determined by the ordinary differential equation
Fϕ ′ x , xϕ
xx ′′ , ...,x ϕ x = A.
n−1 ( n)
3 ◦ . Solution:
w(x, t) = tU (z) + C, z = x/t,
where
C is an arbitrary constant and the function U (z) is determined by the ordinary differential equation
FU z ′ , zU zz ′′ ,
n−1 ...,z ( U n)
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
w − 1 = w(x + C 1 e at , t+C 2 ), where C 1 and C 2 are arbitrary constants, is also a solution of the equation.
2 ◦ . Generalized traveling-wave solution:
w = w(z), − z = x + Ce at ,
where
C is an arbitrary constant and the function w(z) is determined by the ordinary differential equation
F w, w ′ z , w ′′ zz ,
...,w ( n)
+ azw ′ z = 0.
5. =F w, x
,x 2
,...,x n
The substitution x= ✡ e z leads to an equation of the form 11.2.1.2.
,...,x n
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
1 = w(C 1 x, C 1 k t+C 2 ),
where C 1 and C 2 are arbitrary constants, is also a solution of the equation.
2 ◦ . Self-similar solution:
w(x, t) = w(z), 1 z = xt /k ,
where the function w(z) is determined by the ordinary differential equation
kz ( k−1 F w, zw ′ , ...,z n w n)
,...,x n
Passing to the new independent variables
we obtain an equation of the form 11.2.3.6:
∂w
∂w
, ...,z n ∂ w .
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
1 = w(x + C 1 , e λC 1 t+C 2 ),
where C 1 and C 2 are arbitrary constants, is also a solution of the equation.
2 ◦ . Generalized traveling-wave solution:
w(x, t) = w(z), z = λx + ln t,
where the function w(z) is determined by the ordinary differential equation
F w, λw
( n) z ′ , ...,λ w z
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
w 1 = C 1 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) = Ae µt ϕ(x),
where
A and µ are arbitrary constants, and the function ϕ(x) is determined by the ordinary differential equation
F x, ϕ ′ x /ϕ, ϕ ′′
xx /ϕ, . . . , ϕ n) x /ϕ = µ.
For β = 1, see equation 11.2.3.9.
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
w 1 = C 1 w(x, C β−1 1 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) = (1 − β)At + B 1− β ϕ(x),
where
A and B are arbitrary constants, and the function ϕ(x) is determined by the ordinary differential equation
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
1 w 1 = w(x, C 1 t+C 2 )+ ln C 1 , β
where C 1 and C 2 are arbitrary constants, is also a solution of the equation.
2 ◦ . Additive separable solution:
1 w(x, t) = − ln( Aβt + B) + ϕ(x), β
where
A and B are arbitrary constants, and the function ϕ(x) is determined by the ordinary differential equation
2 ∂t ,..., ∂x ∂x ∂x ∂x n
1 ◦ . Additive separable solution:
w(x, t) = At + B + ϕ(x),
where
A and B are arbitrary constants, and the function ϕ(x) is determined by the ordinary differential equation
F x, ϕ
xx ′′ /ϕ ′ x , ...,ϕ x /ϕ ′ x = A.
n)
2 ◦ . Generalized separable solution:
w(x, t) = Ae µt Θ( x) + B,
where
A, B, and µ are arbitrary constants, and the function Θ(x) is determined by the ordinary differential equation
F x, Θ
xx ′′ /Θ ′ x ...,Θ x /Θ ′ x = µΘ.
n)
For β = 1, see equation 11.2.3.12.
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
w 1 = C 1 w(x, C β−1 1 t+C 2 )+ C 3 ,
where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.
2 ◦ . Additive separable solution:
w(x, t) = At + B + ϕ(x),
where
A and B are arbitrary constants, and the function ϕ(x) is determined by the ordinary differential equation
F x, ϕ ′′ xx /ϕ ′ x , ...,ϕ ( n) x /ϕ ′ x = A.
3 ◦ . Generalized separable solution:
w(x, t) = A(1 − β)t + C 1 1− β Θ( x) + B + C 2 ,
where
A, B, C 1 , and C 2 are arbitrary constants, and the function Θ( x) is determined by the ordinary differential equation
F x, Θ ′′ xx /Θ ′ x ...,Θ ( n) x /Θ ′ x = AΘ + AB.
11.2.4. Equations of the Form ∂w n = F x, t, w, ∂w , ...,∂ w ∂t n ∂x ∂x
∂w
∂w ∂ 2 w
2 ∂t ,..., ∂x ∂x ∂x n + g(t)w.
1. =F x, t,
Suppose w(x, t) is a solution of the equation in question. Then the function
w 1 = w(x, t) + C exp
g(t) dt ,
where
C are arbitrary constants, is also a solution of the equation. ∂w
w = w(ξ), ξ = ax + bt,
where the function w(ξ) is determined by the ordinary differential equation
F ξ, w, aw ′ , ...,a n ( w n) − bw ′ ξ ξ ξ = 0.
w = ϕ(ξ) + Ct, ξ = ax + bt,
where
C is an arbitrary constant and the function ϕ(ξ) is determined by the ordinary differential equation
′ , n ( F ξ, aϕ n)
ξ ...,a ϕ ξ − bϕ ξ ′ −
C = 0,
whose order can be reduced with the substitution U (ξ) = ϕ ′ ξ .
4. = f (t)x k Φ w, x
,...,x n ∂ w
+ xg(t)
∂x Passing to the new independent variables
z = xG(t), − τ= f (t)G k ( t) dt, G(t) = exp g(t) dt , one arrives at a simpler equation of the form 11.2.3.6:
, ...,z n
+ f (t)e .
Generalized separable solution:
w(x, t) = e λx
f (t)
E(t) − A+ dt + Be λx E(t),
E(t)
2 t, λ 2 , ...,λ n dt , where
E(t) = exp
A and B are arbitrary constants. ∂w
6. =w Φ t,
+ f (t)e λx + g(t)e – 2 λx n .
Generalized separable solution:
g(t) w(x, t) = e E(t) A+
λx
f (t)
dt + e λx E(t) B+ dt ,
E(t)
E(t)
2 E(t) = exp 2 Φ t, λ , ...,λ n dt , where
A and B are arbitrary constants. ∂w
7. =w Φ t, ∂t
+ f (t) cosh(λx) + g(t) sinh(λx).
∂x 2
w ∂x 2 n
Generalized separable solution:
Z g(t) w(x, t) = cosh(λx)E(t) A+
f (t)
dt + sinh( λx)E(t) B+ dt ,
E(t)
E(t)
2 E(t) = exp 2 t, λ , ...,λ n dt , where
A and B are arbitrary constants. ∂w
8. =w Φ t, ∂t
+ f (t) cos(λx).
∂x 2
w ∂x 2 n
Generalized separable solution:
f (t)
w(x, t) = cos(λx)E(t) A+
dt +
B sin(λx)E(t),
E(t)
2 t, −λ 2 , . . . , (−1) n λ n dt , where
E(t) = exp
A and B are arbitrary constants.
9. =w Φ t,
+ f (t) cos(λx) + g(t) sin(λx).
∂t
w ∂x 2 w ∂x 2 n
Generalized separable solution:
g(t) w(x, t) = cos(λx)E(t) A+
f (t)
dt + sin( λx)E(t) B+
E(t) = exp
2 t, −λ 2 , . . . , (−1) n λ n dt ,
where
A and B are arbitrary constants. ∂w
1 ∂w 1 ∂ 2 w
10. = f (t)w Φ x,
+ g(t)w.
∂t
w ∂x w ∂x 2 w ∂x n
The transformation
G(t) = exp g(t) dt , leads to a simpler equation of the form 11.2.3.10:
w(x, t) = G(t)u(x, τ ), τ=
f (t)G β−1 ( t) dt,
which has a multiplicative separable solution u = ϕ(x)ψ(τ ). ∂w
+ g(t)w + h(t).
Generalized separable solution:
w(x, t) = ϕ(t)Θ(x) + ψ(t),
where the functions ϕ(t) and ψ(t) are determined by the system of first-order ordinary differential equations (
C is an arbitrary constant)
ϕ ′ t = Af (t)ϕ k + g(t)ϕ,
(2) and the function Θ( x) satisfies the nth-order ordinary differential equation
ψ ′ t = g(t)ψ + Bf (t)ϕ k + h(t),
x, Θ xx /Θ x ...,Θ n) x /Θ x ′ = AΘ + B. The general solution of system (1), (2) is given by
, G(t) = exp g(t) dt , ψ(t) = DG(t) + G(t)
ϕ(t) = G(t)
C + A(1 − k)
f (t)G k−1 ( t) dt
1− k
Bf (t)ϕ k ( t) + h(t)
G(t)
where
A, B, C, and D are arbitrary constants. ∂w
12. = f 1 (t)w + f 0 (t)
Φ x,
+g 1 (t)w + g 0 (t).
∂x n ∂x Generalized separable solution:
w(x, t) = ϕ(t)Θ(x) + ψ(t), w(x, t) = ϕ(t)Θ(x) + ψ(t),
C is an arbitrary constant):
ϕ ′ t = Cf 1 ( t)ϕ k+1 + g 1 ( t)ϕ,
(2) and the function Θ( x) satisfies the nth-order ordinary differential equation
ψ ′ t = Cf 1 ( t)ϕ k + g 1 ( t) ψ + Cf 0 ( t)ϕ k + g 0 ( t),
Φ x, Θ ′′ xx /Θ
x ′ , ...,Θ x /Θ ′ x = C.
n)
The general solution of system (1), (2) is given by
−1 /k
ϕ(t) = G(t)
g 1 ( t) dt , ψ(t) = Bϕ(t) + ϕ(t)
A − kC
f 1 ( t)G k ( t) dt
, G(t) = exp
A, B, and C are arbitrary constants. ∂w
∂w ∂ 2 w
13. = f (t)e Φ x,
βw
+ g(t).
The transformation
Z w(x, t) = u(x, τ ) + G(t), τ=
G(t) = g(t) dt, leads to a simpler equation of the form 11.2.3.11:
f (t) exp βG(t) dt,
which has a solution in the additive separable form u = ϕ(x) + ψ(τ ). ∂w
+ f (t)e + g(t)e
There is a generalized separable solution of the form
w(x, t) = e − λx ϕ(t) + e λx ψ(t).
+ f (t) cosh(λx) + g(t) sinh(λx).
There is a generalized separable solution of the form
w(x, t) = cosh(λx)ϕ(t) + sinh(λx)ψ(t).
+ f (t) cos(λx) + g(t) sin(λx).
There is a generalized separable solution of the form
w(x, t) = cos(λx)ϕ(t) + sin(λx)ψ(t).
P n (–1) i+k
, k = 0, 1, . . . , n. Multiplicative separable solution:
17. = wF (t, ζ 0 ,ζ 1 ,...,ζ ),
w(x, t) = (C 0 + C 1 x+···+C n x n ) ϕ(t),
where C 0 , C 1 , ...,C n are arbitrary constants, and the function ϕ = ϕ(t) is determined by the ordinary differential equation
ϕ ☛✂☞ ′ t = ϕF (t, C 0 ϕ, C 1 ϕ, . . . , C n ϕ).
Reference : Ph. W. Doyle (1996), the case ∂ t
F ≡ 0 was treated.