
234 FIRST-ORDER EQUATIONS WITH THREE OR MORE INDEPENDENT VARIABLES
22.
∂w
∂x
+ a
∂w
∂y
+ b
∂w
∂z
= c tanh
n
(βx)w + k coth
m
(λx).
General solution:
w = E(x)Φ(y − ax, z − bx) + kE(x)
Z
coth
m
(λx)
dx
E(x)
,
E(x) = exp
c
Z
tanh
n
(βx) dx
.
23.
∂w
∂x
+ b cosh
n
(βx)
∂w
∂y
+ c sinh
k
(λy)
∂w
∂z
= aw + s cosh
m
(µx).
This is a special case of equation 2.4.7.8 with f(x) = b cosh
n
(βx), g(y) = c sinh
k
(λy),
and h(x) = s cosh
m
(µx).
24.
∂w
∂x
+ a tanh
n
(βx)
∂w
∂y
+ b coth
k
(λx)
∂w
∂z
= cw + s tanh
m
(µx).
This is a special case of equation 2.4.7.1 with f(x) = a tanh
n
(βx), g(x) = b coth
k
(λx),
h(x) = c, and p(x) = s tanh
m
(µx).
25. b
1
sinh
n
1
(λ
1
x)
∂w
∂x
+ b
2
cosh
n
2
(λ
2
y)
∂w
∂y
+ b
3
sinh
n
3
(λ
3
z)
∂w
∂z
= aw + c
1
cosh
k
1
(β
1
x) + c
2
sinh ...