
Chapter 1 1
Higher-Order Partial Differential
Equations
11.1 Third-Order Partial Differential Equations
11.1.1 O n e-Dimensional Equations Containing the First Derivative
in t
1.
∂w
∂t
+
∂
3
w
∂x
3
= Φ(x, t).
Linearized Korteweg–de Vries equation.
1
◦
. Particular solutions of the homogeneous equation with Φ(x, t) = 0:
w(x, t) = a(x
3
− 6t) + bx
2
+ cx + k,
w(x, t) = a(x
5
− 60x
2
t) + b(x
4
− 24xt),
w(x, t) = a sin(λx + λ
3
t) + b cos(λx + λ
3
t) + c,
w(x, t) = a sinh(λx −λ
3
t) + b cosh(λx −λ
3
t) + c,
w(x, t) = exp
−λ
3
t
a exp
λx
+ b exp
−
1
2
λx
sin
√
3
2
λx + c
,
where a, b, c, k, and λ are arbitrary constants.
2
◦
. Solution given by a formal series in powers of t (Φ(x, t) = 0):
w(x,