Where did I make a mistake?
As Daniel said, a_j[t] is interpreted as Pattern[a, Blank[j]][t]
because _ (underscore) has a build-in meaning in Mathematica (look up
Blank). It can't be a part of a symbol name as in other programming
languages. That's probably what's causing your problems. Use Subscript
[a,j][t] instead. As for the power, ^ will work just fine, but you
said you wanted a superscript ;)
Norbert