
204 FIRST-ORDER EQUATIONS WITH THREE OR MORE INDEPENDENT VARIABLES
7. (a
1
+ a
2
e
αx
)
∂w
∂x
+ (b
1
+ b
2
e
βy
)
∂w
∂y
+ (c
1
+ c
2
e
γz
)
∂w
∂z
=
k
1
+ k
2
e
αx
w.
General solution: w = Φ(u
1
, u
2
) exp
k
1
a
1
x +
1
α
k
2
a
2
−
k
1
a
1
ln(a
1
+ a
2
e
αx
)
, where
u
1
=
1
a
1
α
αx − ln(a
1
+ a
2
e
αx
)
−
1
b
1
β
βy − ln(b
1
+ b
2
e
βy
)
,
u
2
=
1
a
1
α
αx − ln(a
1
+ a
2
e
αx
)
−
1
c
1
γ
γz − ln(c
1
+ c
2
e
γz
)
.
8. e
βy
(a
1
+ a
2
e
αx
)
∂w
∂x
+ e
αx
(b
1
+ b
2
e
βy
)
∂w
∂y
+ ce
βy+γz
∂w
∂z
= k
3
e
βy
(k
1
+ k
2
e
αx
)w.
General solution: w = Φ(u
1
, u
2
) exp
k
1
k
3
a
1
x +
k
3
α
k
2
a
2
−
k
1
a
1
ln(a
1
+ a
2
e
αx
)
, where
u
1
=
1
a
2
α
ln(a
1
+a
2
e
αx
)−
1
b
1
β
ln(b
1
+b
2
e
βy
), u
2
=
1
a
1
α
αx−ln(a
1
+a
2
e
αx
)
+
1
cγ
e
−γz
.
◮ Coefficients of equations contain exponential and power-law functions.
9.
∂w
∂x
+ ax
n
∂w
∂y
+ bx
m
∂w
∂z
=
ce
λx
y +