
2.4. Equations of the Form f
1
∂w
∂x
+ f
2
∂w
∂y
+ f
3
∂w
∂z
= f
4
w + f
5
, f
n
= f
n
(x, y, z) 249
1
◦
. Any two out of three functions u, v, and w can be assumed to be arbitrary, and the
remaining function is determined by a simple single integration of Eq. (1).
This method permits one to obtain the general solution in the form
u = u(x, y, z) is an arbitrary function,
v = v(x, y, z) is an arbitrary function,
w = −
Z
(u
x
+ v
y
) dz + ξ
3
(x, y).
2
◦
. Suppose that ψ
(1)
= ψ
(1)
(x, y, z) and ψ
(1)
= ψ
(2)
(x, y, z) are two arbitrary twice
continuously differentiable functions.
2.1. The general solution of Eq. (1) can be represented, for example, in the simple form
u = (u, v, w), u = ψ
(1) ...