
1036 HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS
2
◦
. General solution w ith Φ (x, t) = 0 and a = −α
2
< 0:
w = w
1
+ w
2
,
where w
1
and w
2
are arbitrary solutions of two simpler nth-order equations
∂w
1
∂t
+ α
∂
n
w
1
∂x
n
= 0,
∂w
2
∂t
−α
∂
n
w
2
∂x
n
= 0.
See equation 11.6.5.1 or 11.6.5.2 for information on the solution of these equations de-
pending on whether n is odd or even.
3
◦
. Formal series solution for Φ(x, t) = 0:
w(x, t) =
∞
X
k=0
(−1)
k
a
k
t
2k
(2k)!
d
2nk
f(x)
dx
2nk
+
∞
X
k=0
(−1)
k
a
k
t
2k+ 1
(2k + 1)!
d
2nk
g(x)
dx
2nk
,
where f(x) and g(x) are arbitrary infinitely differentiable functions and d
0
f(x)/dx
0
=
f(x). This solution satisfies the initial conditions w(x, 0) = f (x) and ∂
t
w(x, 0) = g(x).