  In the framework of the PDE's algebraic topology, previously introduced by A.
Pr\'astaro, are considered {\em exotic differential equations}, i.e.,
differential equations admitting Cauchy manifolds $N$ identifiable with exotic
spheres, or such that their boundaries $\partial N$ are exotic spheres. For
such equations are obtained local and global existence theorems and stability
theorems. In particular the smooth ($4$-dimensional) Poincar\'e conjecture is
proved. This allows to complete the previous Theorem 4.59 in \cite{PRA17} also
for the case $n=4$.
