
11.6. Higher-Order Linear Equations with Constant Coefficients 1041
5.
∂
2
∂x
2
+
∂
2
∂y
2
n
w = Φ(x, y), n = 1, 2, . . .
This is a nonhomogeneous polyharmonic equation of order n with two independent vari-
ables.
Particular solution:
w(x, y) =
1
k
n
Z
∞
−∞
Z
∞
−∞
Φ(ξ, η)[(x − ξ)
2
+ (y − η)
2
]
n−1
ln[(x − ξ)
2
+ (y − η)
2
] dξ dη,
k
n
= π2
2n
[(n −1)!]
2
.
The general solution is given by the sum of any particular solution of the nonhomogeneous
equation and the general solution of the homogeneous equation (see equation 11.6.7.4,
Item 1
◦
).
6.
m
X
k=0
a
k
∆
k
w = 0, ∆ ≡
∂
2
∂x
2
+
∂
2
∂y
2
.
Particular solutions:
w(x, y) =
m
X
n=1
u
n
(x, y),
where the u
n
are solutions of the Helmholtz equations ∆u
n
− λ
n
u
n
= 0 and ...