2.2. SECOND-ORDER PDES 37
where S satisfies S
00
C kS D 0. We will distinguish between two different cases:
.i/ k D 0;
.i i/ k ¤ 0:
In the case of k D 0, then S D c
1
x C c
2
. However, we can set c
1
D 1 and c
2
= 0 without
loss of generality. In this case, (2.210a) becomes
u
t
3
x
u
x
D
3
x
2
u; (2.211)
which is solved giving
u D xF .x
2
C 6t/; (2.212)
where F is an arbitrary function of its argument. Substitution of (2.212) into (2.210b) (with
k D 0) gives
F
00
C
q
4
F
3
D 0; (2.213)
where F D F ./; D x
2
C 6t. e solution of (2.213) can be expressed in terms of the Jacobi
elliptic functions so
F D
2
p
q
cn
;
1
p
2
and F D
2
p
q
nc
;
1
p
2
depending on the sign of q.
In the case of k ¤ 0, we integrate (2.210a), giving
u D
kS.x/
S
0
.x/
F
3kt ln.S
0
/
; (2.214)
where F is again an arbitrary function of its argument; substitution into (2.210b) gives rise to
the ODE
F
00
C 3F
0
C 2F C qF
3
D 0; (2.215)
where F D F ./; D 3kt ln.S
0
/. If we introduce the change of variables
D ln ; F D G; (2.216)
where G D G./, then (2.215) becomes
G
00
C qG
3
D 0; (2.217)
which is essentially (2.213). If we impose the change of variables (2.216) on the solution (2.214),
then we obtain
u D kS.x/e
3kt
G
S
0
.x/e
3kt
; (2.218)