
174 FIRST-ORDER EQUATIONS WITH THREE OR MORE INDEPENDENT VARIABLES
17. ax
2
∂w
∂x
+ by
2
∂w
∂y
+ cz
2
∂w
∂z
= kxy.
General solution: w =
kxy
ax − by
ln
ax
y
+ Φ
b
x
−
a
y
,
c
x
−
a
z
.
18. ax
2
∂w
∂x
+ by
2
∂w
∂y
+ cz
2
∂w
∂z
= αx
2
+ βy
2
+ γz
2
.
General solution: w =
α
a
x +
β
b
y +
γ
c
z + Φ
b
x
−
a
y
,
c
x
−
a
z
.
◮ Coefficients of equations contain other p owers in x, y, and z.
19.
∂w
∂x
+ a
∂w
∂y
+ b
∂w
∂z
= xyz.
General solution: w =
1
2
x
2
yz −
1
6
x
3
(az + by) +
1
12
abx
4
+ Φ(y − ax, z − bx).
20. a
∂w
∂x
+ b
∂w
∂y
+ c
∂w
∂z
= kx
3
+ sy
2
.
General solution: w =
k
4a
x
4
+
s
3a
y
3
+ Φ(bx − ay, cx − az).
21. a
∂w
∂x
+ by
∂w
∂y
+ cz
∂w
∂z
= kx + s
√
x.
General solution: w =
k
2a
x
2
+
2s
3a
x
3/2
+ Φ
|y|
a
e
−bx
, |z|
a
e
−cx
.
22.
∂w
∂x
+ az
∂w
∂y
+ by
∂w
∂z
= c
√
x + s.
General soluti ...