  Computing analytic B\'ezout identities remains a difficult task, which has
many applications in control theory. Flat PDE systems have cast a new light on
this problem. We consider here a simple case of special interest: a rod of
length $a+b$, insulated at both ends and heated at point $x=a$. The case $a=0$
is classical, the temperature of the other end $\theta(b,t)$ being then a flat
output, with parametrization $\theta(x,t)=\cosh((b-x)(\partial/\partial
t)^{1/2}\theta(b,t)$.
  When $a$ and $b$ are integers, with $a$ odd and $b$ even, the system is flat
and the flat output is obtained from the B\'ezout identity
$f(x)\cosh(ax)+g(x)\cosh(bx)=1$, the omputation of which boils down to a
B\'ezout identity of Chebyshev polynomials. But this form is not the most
efficient and a smaller expression $f(x)=\sum_{k=1}^{n} c_{k}\cosh(kx)$ may be
computed in linear time.
  These results are compared with an approximations by a finite system, using a
classical discretization.
  We provide experimental computations, approximating a non rational value $r$
by a sequence of fractions $b/a$, showing that the power series for the
B\'ezout relation seems to converge.
