144 6. FUNCTIONAL SEPARABILITY
If ¤ 0 then A, B, and C are at most quadratic, so from (6.144) and (6.113), solutions are of
the form
1
f .u/ D
x
2
C y
2
C z
2
C k; (6.147)
k is constant and it’s relatively straightforward to deduce that (6.112) will admit solutions of
this type, provided that Q is of the form
Q D
4kf
00
f
03
2.2ff
00
3f
02
/
f
03
(6.148)
6.1 EXERCISES
6.1. Seek separable solutions for the following:
.i/ u
t
D
u
x
p
u
x
u D
.
a.x/t C b.x/
/
2
.i i/ u
t
D
u
x
p
u
x
C
u
y
p
u
y
u D
.
a.x; y/t C b.x; y/
/
2
6.2. Determine the form of Q such that
u
xx
C u
yy
D Q.u/ (6.149)
admits functional separable solutions of the form ([78], [79], and [80])
f .u/ D A.t/ C B.x/: (6.150)
6.3. Determine the form of Q such that
u
tt
D u
xx
C Q.u; u
t
; u
x
/ (6.151)
admits solutions of the form [83]
u D A.t / C B.x/: (6.152)
6.4. Determine the form of Q such that
u
t
D u
xx
C Q.u; ux/ (6.153)
admits solutions of the form
f .u/ D A.t/ C B.x/: (6.154)
1
we have omitted the linear terms in x, y and z as the original PDE admits translation in these variables.
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