We consider the two-loop self-mass sunrise amplitude with two equal
masses $M$ and the external invariant equal to the square of the
third mass $m$ in the usual $d$-continuous dimensional regularization.
We write a second order differential equation for the amplitude
in $x=m/M$ and show as solve it in close analytic form. As a result,
all the coefficients of the Laurent expansion in $(d-4)$
of the amplitude are expressed in terms of harmonic polylogarithms of
argument $x$ and increasing weight. As a by product, we give
the explicit analytic expressions of the value of the amplitude at
$x=1$, corresponding to the on-mass-shell sunrise amplitude in the
equal mass case, up to the $(d-4)^5$ term included.
