  We address the enumeration of q-coloured planar maps counted bythe number of
edges and the number of monochromatic edges. We prove that the associated
generating function is differentially algebraic,that is, satisfies a
non-trivial polynomial differential equation withrespect to the edge variable.
We give explicitly a differential systemthat characterizes this series. We then
prove a similar result for planar triangulations, thus generalizing a result of
Tutte dealing with their proper q-colourings. Instatistical physics terms, we
solvethe q-state Potts model on random planar lattices. This work follows a
first paper by the same authors, where the generating functionwas proved to be
algebraic for certain values of q,including q=1, 2 and 3. It isknown to be
transcendental in general. In contrast, our differential system holds for an
indeterminate q.For certain special cases of combinatorial interest (four
colours; properq-colourings; maps equipped with a spanning forest), we derive
from this system, in the case of triangulations, an explicit differential
equation of order 2 defining the generating function. For general planar maps,
we also obtain a differential equation of order 3 for the four-colour case and
for the self-dual Potts model.
