
254 FIRST-ORDER EQUATIONS WITH THREE OR MORE INDEPENDENT VARIABLES
15.
∂w
∂x
1
+
n
X
k=2
a
k
x
k
+ b
k
x
m
k
1
∂w
∂x
k
= c
1
w + c
2
.
This is a special case of equation 2.4.9.33 with f
k
(x
1
) = b
k
x
m
k
1
, g(x
1
) = c
1
, and h(x
1
) = c
2
.
16.
n
X
k=1
a
k
x
m
k
k
∂w
∂x
k
= bw +
n
X
k=1
c
k
x
s
k
k
.
This is a special case of equation 2.4.9.34 with f
k
(x
k
) = a
k
x
m
k
k
and g
k
(x
k
) = c
k
x
s
k
k
.
17.
∂w
∂x
1
+
n
X
k=2
b
k
x
m
k
k−1
x
k
+ c
k
x
a
k
k
∂w
∂x
k
= s
1
w + s
2
.
This is a special case of equation 2.4.9.36 with f
k
(. . .)=b
k
x
m
k
k−1
, g
k
(. . .)=c
k
, h
1
(. . .)=s
1
,
and h
2
(. . .) = s
2
.
◮ Other equations containing arbitrary parameters.
18.
∂w
∂x
1
+
n
X
k=2
a
k
e
λ
k
x
1
∂w
∂x
k
= bw + c.
This is a special case of equation 2.4.9.32 with f
k
(x
1
) = a
k
e
λ
k
x
1
, g(