
1.4. Equations of the Form f(x, y)
∂w
∂x
+ g(x, y)
∂w
∂y
= h
1
(x, y)w + h
0
(x, y) 133
18. a
∂w
∂x
+ b
∂w
∂y
=
c
1
arccot(λ
1
x) + c
2
arccot(λ
2
y)
w
+ s
1
arccot
n
(β
1
x) + s
2
arccot
k
(β
2
y).
This is a special case of equation 1.4.7.16 in which f(x) = c
1
arccot(λ
1
x), g(y) =
c
2
arccot(λ
2
y), p(x) = s
1
arccot
n
(β
1
x), and q(y) = s
2
arccot
k
(β
2
y).
19. a arccot
n
(λx)
∂w
∂x
+ b arccot
m
(µx)
∂w
∂y
= c arccot
k
(νx)w + p arccot
s
(βy).
This is a special case of equation 1.4.7.22 with f(x) = a arccot
n
(λx), g
1
(x) ≡ 0, g
0
(x) =
b arccot
m
(µx), h(x, y) = c arccot
k
(νx), and F (x, y) = p arccot
s
(βy).
20. a arccot
n
(λx)
∂w
∂x
+ b arccot
m
(µx)
∂w
∂y
= c arccot
k
(νy)w + p arccot
s
(βx).
This is a special case of equation 1.4.7.22 ...