
2.3. Equations of the Form f
1
∂w
∂x
+ f
2
∂w
∂y
+ f
3
∂w
∂z
= f
4
w, f
n
= f
n
(x, y, z) 199
13.
∂w
∂x
+ (a
2
xy + a
1
x
2
+ a
0
)
∂w
∂y
+ (b
2
xy + b
1
x
2
+ b
0
)
∂w
∂z
= (c
2
y + c
1
z + c
0
x + s)w.
This is a special case of equation 2.3.7.4 with f
1
(x) = a
2
x, f
2
(x) = a
1
x
2
+a
0
, g
1
(x) = b
2
x,
g
2
(x) = b
1
x
2
+ b
0
, h
2
(x) = c
2
, h
1
(x) = c
1
, and h
0
(x) = c
0
x + s.
14. ax
∂w
∂x
+ by
∂w
∂y
+ cz
∂w
∂z
= x( λx + βy + γz)w.
1
◦
. General solution for b 6= −a and c 6= −a:
w = exp
λ
2a
x
2
+
β
a + b
xy +
γ
a + c
xz
Φ
x|y|
−a/b
, x|z|
−a/c
.
2
◦
. General solution for b = −a and c 6= −a:
w = exp
1
2a
x
λx + 2βy ln |x|
+
γ
a + c
xz
Φ
xy, x|z|
−a/c
.
3
◦
. General solution for b = c = −a:
w = exp
1
2a
x
λx + 2(βy + γz) ln |x|
Φ(xy, xz).
15. ax
2
∂w
∂x
+ bxy ...