
2.1. Equations of the Form f(x, y, z)
∂w
∂x
+ g(x, y, z)
∂w
∂y
+ h(x, y, z)
∂w
∂z
= 0 169
22. ax
∂w
∂x
+ by
∂w
∂y
+ f (x, y)g(z)
∂w
∂z
= 0.
Integral basis:
u
1
= x
b
y
−a
, u
2
= b
Z
dz
g(z)
−
Z
y
y
0
t
−1
f
xy
−a/b
t
a/b
, t
dt.
23.
∂w
∂x
+
f
1
(x)y + f
2
(x)
∂w
∂y
+
g(x, y)z + h(x, y)
∂w
∂z
= 0.
Integral basis:
u
1
= yF (x) −
Z
f
2
(x)F (x) dx, F (x) = exp
−
Z
f
1
(x) dx
,
u
2
= zG(x, u
1
) −
Z
x
x
0
¯
h(t, u
1
)G(t, u
1
) dt, G(x, u
1
) = exp
−
Z
x
x
0
¯g(t, u
1
) dt
.
Here ¯g(x, u
1
) ≡ g(x, y) and
¯
h(x, u
1
) ≡ h(x, y) (y is expressed via x and u
1
from the first
integral), and x
0
is an arbitrary number.
⊙ Literature: A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux (2002).
24.
∂w
∂x
+
f
1
(x)y + f
2
(x)y
k
∂w
∂y
+
g(x, y)z + h(x, y)z
m
∂w
∂z