
846 SECOND-ORDER ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES
4
◦
. The transformation (specified by A. I. Zhurov, private communication, 2001)
x
2−n
2
= Ar cos θ, y
2−m
2
= Br sin θ,
where A
2
= a(2 −n)
2
and B
2
= b(2 −m)
2
, leads to the equation
∂
2
w
∂r
2
+
4−nm
(2−n)(2−m)
1
r
∂w
∂r
+
1
r
2
∂
2
w
∂θ
2
−
2
r
2
(nm−n−m) cos 2θ+(n−m)
(2−n)(2−m) sin 2θ
∂w
∂θ
=4cw,
which admits separable solutions of the form w(r, θ) = F
1
(r)F
2
(θ).
4.
∂
∂x
hh
a(x + k)
n
∂w
∂x
ii
+
∂
∂y
hh
b(y + s)
m
∂w
∂y
ii
= c.
The transformation ζ = x + k, η = y + s leads to an equation of the form 9.4.3.2:
∂
∂ζ
aζ
n
∂w
∂ζ
+
∂
∂η
bη
m
∂w
∂η
= c.
5.
∂
∂x
hh
a(x + k)
n
∂w
∂x
ii
+
∂
∂y
hh
b(y + s)
m
∂w
∂y
ii
= cw.
The transformation ζ = x + k, η = y + s leads to an equation