  (Quoted from the article) This article pursue the series, initiated by
[Ric14], dedicated to Pulita's {\pi}-exponentials and p-adic differential
equation of rank one with coefficients a polynomial in a ultrametric extension
of the field of $p$-adic numbers. We complement [Ric14] with a closed formula
for the index. The "$p$-typical" particular case answers one problem studied in
[Mor10]. We also answer a question [Rob86, {\S}2.4] of Robba on the comparison
from rational cohomology toward Dwork cohomology. We even indicate a procedure
to palliate the lack of isomorphy of this comparison. We establish by the way a
characterisation of soluble equations up to equivalence on the dagger algebra.
An appendix determine the polynomial complexity of the derived algorithm.
