
11.2. Fourth-Order One-Dimensional Nonstationary Equations 961
◮ Particular solutions and the formal series solution.
1
◦
. Particular solutions:
w(x, t) = (Ax
3
+ Bx
2
+ Cx + D)t + A
1
x
3
+ B
1
x
2
+ C
1
x + D
1
,
w(x, t) =
A sin(λx) + B cos(λx) + C sinh(λx) + D cosh(λx)
sin(λ
2
at),
w(x, t) =
A sin(λx) + B cos(λx) + C sinh(λx) + D cosh(λx)
cos(λ
2
at),
w(x, t) = exp(−λx)
A sin(λx) + B cos(λx)
C exp(−2λ
2
at) + D exp(2λ
2
at)
,
w(x, t) = exp(λx)
A sin(λx) + B cos(λx)
C exp(−2λ
2
at) + D exp(2λ
2
at)
,
where A, B, C, D, A
1
, B
1
, C
1
, D
1
, and λ are arbitrary constants.
2
◦
. Formal series solution:
w(x, t) =
∞
X
k=0
(−1)
k
a
2k
t
2k
(2k)!
d
4k
f(x)
dx
4k
+
∞
X
k=0
(−1)
k
a
2k
t
2k+ 1
(2k + 1)!
d
4k
g(x)
dx
4k
,
where ...