
2.2. Equations of the Form f
1
∂w
∂x
+ f
2
∂w
∂y
+ f
3
∂w
∂z
= f
4
, f
n
= f
n
(x, y, z) 179
14.
∂w
∂x
+
ae
αx
y
2
+be
−αx
∂w
∂y
+
de
βx
z
2
+ce
γx
(γ −cde
(β+γ)x
)
∂w
∂z
= ke
λx
.
General solution: w =
k
λ
e
λx
+ Φ(u
1
, u
2
), where u
1
, u
2
is an integral basis of the homo-
geneous equation 2.1.2.15.
15.
∂w
∂x
+
a
1
e
λ
1
x
y + b
1
e
β
1
x
y
k
∂w
∂y
+
a
2
e
λ
2
x
z + b
2
e
β
2
x
z
m
∂w
∂z
= cx
s
.
This is a special case of equation 2.2.7.7 with f
1
(x) = a
1
e
λ
1
x
, f
2
(x) = b
1
e
β
1
x
, g
1
(x) =
a
2
e
λ
2
x
, g
2
(x) = b
2
e
β
2
x
, and h(x) = cx
s
.
16.
∂w
∂x
+
a
1
e
β
1
x
y + b
1
e
γ
1
x
y
k
∂w
∂y
+
a
2
e
β
2
x
+ b
2
e
γ
2
x+λz
∂w
∂z
= cx
s
.
This is a special case of equation 2.2.7.8 with f
1
(x) = a
1
e
β
1
x
, f
2
(x) = b
1
e
γ
1
x
, g
1
(x) =
a
2
e
β
2
x
, g
2
(x) = b
2
e
γ
2
x
, and h(x) = cx
s
.
17.
∂w
∂x
+