
144 FIRST-ORDER EQUATIONS WITH THREE OR MORE INDEPENDENT VARIABLES
26. a
∂w
∂x
+ xz
∂w
∂y
− xy
∂w
∂z
= 0.
Integral basis:
u
1
= y
2
+ z
2
, u
2
= y sin
x
2
2a
+ z cos
x
2
2a
.
The function u
2
= x
2
+ 2a arctan(z/y) can also be taken to be the second integral.
⊙ Literature: E. Kamke (1965).
27. cx
∂w
∂x
+ cy
∂w
∂y
+ (ax
2
+ by
2
)
∂w
∂z
= 0.
Integral basis: u
1
= ax
2
+ by
2
− 2cz, u
2
= y/x.
28. cz
∂w
∂x
− a(2ax − b)y
∂w
∂y
+ a(2ax − b)z
∂w
∂z
= 0.
Integral basis: u
1
= yz, u
2
= ax(ax −b) − cz.
29. acx
2
∂w
∂x
− acxy
∂w
∂y
− b
2
y
2
∂w
∂z
= 0.
Integral basis: u
1
= xy, u
2
= 3acxyz − b
2
y
3
.
30. ax
2
∂w
∂x
+ by
2
∂w
∂y
+ cz
2
∂w
∂z
= 0.
Any two of the functions
u
1
=
1
by
−
1
ax
, u
2
=
1
cz
−
1
by
, u
3
=
1
ax
−
1
cz
form an integral basis.
⊙ Literature: E. Kamke