
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) 209
26.
∂w
∂x
+ a tanh
n
(λx)
∂w
∂y
+ b coth
m
(βx)
∂w
∂z
= s coth
k
(γx)w.
This is a special case of equation 2.3.7.1 with f(x) = a tanh
n
(λx), g(x) = b coth
m
(βx),
h
2
(x) = h
1
(x) = 0, and h
0
(x) = s coth
k
(γx).
27. a sinh(λx)
∂w
∂x
+ b sinh(βy)
∂w
∂y
+ c sinh(γz)
∂w
∂z
= k cosh(λx)w.
General solution: w =
sinh(λx)
k/aλ
Φ(u
1
, u
2
), where
u
1
=
1
aλ
ln
tanh
λx
2
−
1
bβ
ln
tanh
βy
2
, u
2
=
1
aλ
ln
tanh
λx
2
−
1
cγ
ln
tanh
γz
2
.
2.3.4 Equations Containing Logarithmic Functions
◮ Coefficients of equations contain logarithmic functions.
1.
∂w
∂x
+ a
∂w
∂y
+ b
∂w
∂z
= c ln
n
(βx)w.
General solution: w = Φ(y − ax, z − bx) exp
c
Z
ln