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This means, if we assume that X.x/
¤
0 for x
2
.0; 1/ and T.t/
¤
0 for t>0,that
T
0
.t /
T.t/
X
00
.x/
X.x/
D
for all x
2
.0; 1/; t > 0:
(8.48)
The left-hand side of this equation only depends on time t and the right-hand
side only depends on the space coordinate x. This is interesting, and very important!
What does it mean? Well, it means that if a function
u
of the form (8.46) satisfies
(
8.44
)and(
8.45
), then there must exist a constant such that
T
0
.t /
T.t/
D
;
(8.49)
X
00
.x/
X.x/
D
:
(8.50)
It is important that you understand this. Take a few minutes to think about it! You
mayalsowanttodoExercise
8.11
.
Note that (
8.49
)and(
8.50
) are two ordinary differential equations.
17
In these two
equations there are three unknowns; two functions T , X and a constant . Indicating
that there might be more than one solution to this system. This is indeed the case.
There are infinitely many solutions!
We have already encountered (
8.49
) in Chap. 2 in connection with models for
exponential growth of rabbit populations. Its solution, see Chap. 2, is given in terms
of the exponential function
T.t/
D
ce
t
;
(8.51)
where c is a constant. In this case there is no initial condition present. Thus, this
function satisfies (
8.49
) for all constants c.
Next, we turn our attention toward (
8.50
). This equation can also be written on
the form
X
00
.x/
D
X.x/;
(8.52)
with the boundary conditions
X.0/
D
0 and X.1/
D
0;
(8.53)
see (8.47). Thus, we are seeking a function such that the second-order derivative of
this function equals the function itself multiplied by a constant. From introductory
courses in calculus we know that
sin
0
.x/
D
cos.x/;
cos
0
.x/
D
sin.x/;
17
There are no partial derivatives present!
