
154 FIRST-ORDER EQUATIONS WITH THREE OR MORE INDEPENDENT VARIABLES
22. (ax
n
e
λy
+ bxy
m
)
∂w
∂x
+ e
µy
∂w
∂y
+ (cy
l
z
k
+ dy
p
z)
∂w
∂z
= 0.
Integral basis:
u
1
= x
1−n
E
1
+ a(n − 1)
Z
e
(λ−µ)y
E
1
dy, E
1
= exp
b(n −1)
Z
y
m
e
−µy
dy
,
u
2
= z
1−k
E
2
+ c(k − 1)
Z
y
l
e
−µy
E
2
dy, E
2
= exp
d(k − 1)
Z
y
p
e
−µy
dy
.
23.
∂w
∂x
+
y
2
+ 2aαe
αx
2
−a
2
e
2αx
2
∂w
∂y
+
ce
−2βx
2
z
2
+ 2βxz +b
2
c
∂w
∂z
= 0.
Integral basis:
u
1
=
E
y − ae
αx
2
+
Z
E dx, E = exp
2a
Z
e
αx
2
dx
,
u
2
= arctan
1
b
e
−βx
2
z
− bc
Z
e
−βx
2
dx.
24.
∂w
∂x
+
ae
−2αx
2
y
2
+ 2αxy + ab
2
∂w
∂y
+
cx
β
z
2
+ 2γxz + cd
2
x
β
e
2γx
2
∂w
∂z
= 0.
Integral basis:
u
1
= arctan
1
b
e
−αx
2
y
−ab
Z
e
−αx
2
dx, u
2
= arctan
1
d
e
−γx
2
z
−cd
Z
x
β
e
γx
2
dx.
2.1.3 Equations Containing Hyperbolic Fun ctions
◮ Coefficients of equations contain ...