
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) 239
20.
∂w
∂x
+ a cot
n
(βx)
∂w
∂y
+ b cot
k
(λx)
∂w
∂z
= cw + s cot
m
(µx).
This is a special case of equation 2.4.7.1 with f(x) = a cot
n
(βx), g(x) = b cot
k
(λx),
h(x) = c, and p(x) = s cot
m
(µx).
21.
∂w
∂x
+ b cot
n
(βx)
∂w
∂y
+ c cot
k
(λy)
∂w
∂z
= aw + s cot
m
(µx).
This is a special case of equation 2.4.7.8 with f(x) = b cot
n
(βx), g(y) = c cot
k
(λy), and
h(x) = s cot
m
(µx).
22. a
∂w
∂x
+ b cot(βy)
∂w
∂y
+ c cot(γz)
∂w
∂z
= p cot(µx)w + q cot(λx).
General solution: w=
sin(µx)
p/aµ
Φ(u
1
, u
2
)+
q
a
Z
sin(µx)
−p/aµ
cot(λx) dx
, where
u
1
= bβx + a ln
cos(βy)
and u
2
= cγx + a ln
cos(γz)
.
23. a
∂w
∂x
+ b cot(βy)
∂w
∂y
+ c