
8.5. Other Equations with Three Space Variables 765
8.5 Other Equations with Three Space Variables
8.5.1 Equations Containing Arbitrary Parameters
1.
∂
2
w
∂t
2
=
∂
∂x
ax
n
∂w
∂x
+
∂
∂y
by
m
∂w
∂y
+
∂
∂z
cz
k
∂w
∂z
.
This equation admits separable solutions. In addition, for n 6= 2, m 6= 2, and k 6= 2, there
are particular solutions of the form
w = w(ξ, t), ξ
2
= 4
x
2−n
a(2 − n)
2
+
y
2−m
b(2 − m)
2
+
z
2−k
c(2 − k)
2
,
where w(ξ, t) is determined by the one-dimensional nonstationary equation
∂
2
w
∂t
2
=
∂
2
w
∂ξ
2
+
A
ξ
∂w
∂ξ
, A = 2
1
2 −n
+
1
2 −m
+
1
2 −k
− 1.
2.
∂
2
w
∂t
2
+ k
∂w
∂t
= a
2
∂
2
w
∂x
2
+
∂
2
w
∂y
2
+
∂
2
w
∂z
2
+ b
1
∂w
∂x
+ b
2
∂w
∂y
+ b
3
∂w
∂z
+ cw.
The transformation
w(x, y, z, t) = u(x, y, z, τ) exp
−
1
2
kt −
b
1
x +