  We address the Poincar\'e-Perron's classical problem of approximation for
high order linear differential equations in the class of almost periodic type
functions, extending the results for a second order linear differential
equation in [23]. We obtain explicit formulae for solutions of these equations,
for any fixed order $n\ge 3$, by studying a Riccati type equation associated
with the logarithmic derivative of a solution. Moreover, we provide sufficient
conditions to ensure the existence of a fundamental system of solutions. The
fixed point Banach argument allows us to find almost periodic and
asymptotically almost periodic solutions to this Riccati type equation. A
decomposition property of the perturbations induces a decomposition on the
Riccati type equation and its solutions. In particular, by using this
decomposition we obtain asymptotically almost periodic and also $p$-almost
periodic solutions to the Riccati type equation. We illustrate our results for
a third order linear differential equation.
