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As far as I know, solve only works with variables. So as a preliminary step you can substitute some new variables for I.diff(t) and uC.diff(t) and then solve. The other problem occuring is that to solve a system of equations for multiple variables, those variables should be specified as a list. Try this: (first substitute new variables DI and DuC for the two derivatives and then call solve)

sage: E2 = [ e.subs({I.diff(t): DI, uC.diff(t): DuC}) for e in equations ]
sage: E2
[t |--> U(t) == 0.100000000000000*DI + 6*I(t) + uC(t), t |--> I(t) == 1/10000*DuC]
sage: solve(E2, [DI, DuC])
[[DI == 10*U(t) - 60*I(t) - 10*uC(t), DuC == 10000*I(t)]]