
2.1. Equations of the Form f(x, y, z)
∂w
∂x
+ g(x, y, z)
∂w
∂y
+ h(x, y, z)
∂w
∂z
= 0 149
3
◦
. Integral basis for m 6= 1 and n = −1:
u
1
=
1
k
y
k
− a ln |x|, k = 1 −m;
u
2
=
z
1−λ
1 −λ
− b
Z
x
ν
ku
1
+ ak ln |x|
µ/k
dx if λ 6= 1,
ln |z| − b
Z
x
ν
ku
1
+ ak ln |x|
µ/k
dx if λ = 1.
In the integration, u
1
is considered a parameter.
4
◦
. Integral basis for m = 1 and n = −1:
u
1
= x
−a
y, u
2
=
z
1−λ
1 −λ
−
bx
ν+1
y
µ
aµ + ν + 1
if λ 6= 1, aµ + ν 6= −1;
z
1−λ
1 −λ
− by
µ
x
−aµ
ln |x| if λ 6= 1, aµ + ν = −1;
ln |z| −
bx
ν+1
y
µ
aµ + ν + 1
if λ = 1, aµ + ν 6= −1;
ln |z| − by
µ
x
−aµ
ln |x| if λ = 1, aµ + ν = −1.
65.
∂w
∂x
+
a
1
x
n
1
y + b
1
x
m
1
∂w
∂y
+ (a
2
x
n
2
y + b
2
x
m
2
∂w
∂z
= 0.
This is a special case of equation ...