
11.6. Higher-Order Linear Equations with Constant Coefficients 1031
2.
∂w
∂t
+ a
∂
2n+1
w
∂x
2n+1
= Φ(x, t), n = 1, 2, . . . .
1
◦
. Particular solutions of the homogeneous equation with Φ ≡ 0:
w(x, t) = b
x
2n+1
(2n + 1)!
−at
+
2n
X
k=0
c
k
x
k
,
w(x, t) = b
2x
2n+3
(2n + 3)!
−ax
2
t
+ c
x
2n+2
(2n + 2)!
−axt
,
w(x, t) = b exp(λx −aλ
2n+1
t) + c exp(−λx + aλ
2n+1
t),
w(x, t) = b sin
λx + (−1)
n+1
aλ
2n+1
t
+ c cos
λx + (−1)
n+1
aλ
2n+1
t
,
w(x, t) = exp
−aλ
2n+1
t
b exp
λx
+
n
X
k=1
exp(λx cos β
k
)[b
k
cos(λx sin β
k
) + c
k
sin(λx sin β
k
)]
, β
k
=
2πk
2n + 1
,
where b, b
k
, c, c
k
, and λ are arbitrary constants.
2
◦
. Formal series solution for Φ ≡ 0:
w(x, t) = f(x) +
∞
X
k=1
(−1)
k
a
k
t
k
k!
d
2nk+k
f(x)
dx
2nk+k
,
where f(x