
224 FIRST-ORDER EQUATIONS WITH THREE OR MORE INDEPENDENT VARIABLES
8. ax
∂w
∂x
+ by
∂w
∂y
+ cz
∂w
∂z
= (αx + β)w + px + q.
General solution: w =
1
a
x
β/a
e
αx/a
Φ
x
b
y
−a
, x
c
z
−a
+
Z
(px+q)x
−(a+β)/a
e
−αx/a
dx
.
9. x
∂w
∂x
+ az
∂w
∂y
+ by
∂w
∂z
= (cx + k)w + px + q.
General solution: w = x
k
e
cx
Φ(u
1
, u
2
) +
Z
(px + q)x
−k−1
e
−cx
dx
, where
u
1
= by
2
− az
2
, u
2
=
x
√
ab
(by −
√
ab z) if ab > 0,
x
√
−ab
exp
−arctan
√
−ab z
by
if ab < 0.
◮ Coefficients of equations are quadratic in x, y, and z.
10.
∂w
∂x
+ (a
1
x
2
+ a
0
)
∂w
∂y
+ (b
1
x
2
+ b
0
)
∂w
∂z
= (c
1
x + c
0
)w + s
1
x
2
+ s
0
.
This is a special case of equation 2.4.7.1 with f (x) = a
1
x
2
+ a
0
, g(x) = b
1
x
2
+ b
0
,
h(x) = c
1
x + c
0
, and p(x) = s
1
x
2
+ s
0
.
11.
∂w
∂x
+ (b
1
x
2
+ b
0
)
∂w
∂y
+ (