
184 FIRST-ORDER EQUATIONS WITH THREE OR MORE INDEPENDENT VARIABLES
5. a
1
ln
n
1
(λ
1
x)
∂w
∂x
+ b
1
ln
m
1
(β
1
y)
∂w
∂y
+ c
1
ln
k
1
(γ
1
z)
∂w
∂z
= a
2
ln
n
2
(λ
2
x) + b
2
ln
m
2
(β
2
y) + c
2
ln
k
2
(γ
2
z).
This is a special case of equation 2.2.7.10 with f (x) = a
1
ln
n
1
(λ
1
x), g(y) = b
1
ln
m
1
(β
1
y),
h(z) = c
1
ln
k
1
(γ
1
z), ϕ(x) = a
2
ln
n
2
(λ
2
x), ψ(y) = b
2
ln
m
2
(β
2
y), and χ(z) = c
2
ln
k
2
(γ
2
z).
◮ Coefficients of equations contain logarithmic and power-law functions.
6. a
∂w
∂x
+ b
∂w
∂y
+ cx
n
ln
k
(λy)
∂w
∂z
= sy
m
ln
l
(βx).
This is a special case of equation 2.2.7.18 with f (x, y) = cx
n
ln
k
(λy), and g(x, y) =
sy
m
ln
l
(βx).
7.
∂w
∂x
+ ax
n
∂w
∂y
+ bx
m
∂w
∂z
= cy ln
k
(λx) + sz ln
l
(βx).
This is a special case of equation 2.2.7.1 with f(x) =