
2.4. Equations of the Form f
1
∂w
∂x
+ f
2
∂w
∂y
+ f
3
∂w
∂z
= f
4
w + f
5
, f
n
= f
n
(x, y, z) 229
4.
∂w
∂x
+ ae
βx
∂w
∂y
+ be
λz
∂w
∂z
= cw + ke
γx
.
General solution:
w =
e
cx
Φ(u
1
, u
2
) +
k
γ − c
e
γx
if c 6= γ,
e
γx
Φ(u
1
, u
2
) + kx
if c = γ,
where u
1
= ae
βx
−βy and u
2
= bλx + e
−λz
.
5.
∂w
∂x
+
a
1
e
σx
+ a
2
e
λy
∂w
∂y
+
b
1
e
µy
+ b
2
e
βz
∂w
∂z
= c
1
w + c
2
e
νx
.
This is a special case of equation 2.4.7.24 with f
1
(x) = a
1
e
σx
, f
2
(x) = a
2
, g
1
(x, y) = b
1
e
µy
,
g
2
(x, y) = b
2
, h
1
(x, y, z) = c
1
, and h
2
(x, y, z) = c
2
e
νx
.
6. b
1
e
λ
1
x
∂w
∂x
+ b
2
e
λ
2
y
∂w
∂y
+ b
3
e
λ
3
z
∂w
∂z
= aw + c
1
e
β
1
x
+ c
2
e
β
2
y
+ c
3
e
β
3
z
.
This is a special case of equation 2.4.7.16 with f
1
(x) = b
1
e
λ
1
x
, f
2
(y) = b
2
e
λ
2
y
, f
3
(z) =
b
3
e
λ
3
z
, g
1
(x) = c
1
e
β
1
x
, g
2
(y) = c
2
e
β
2
y
, and ...