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.