How to substitute differential equation into another?
You could define:
x[0] = 1;
x'[0] = 2;
Derivative[n_?(GreaterEqualThan[2])][x] = Derivative[n-1][x][#] - Derivative[n-2][x][#]&;
Then:
x'''[t]
x'''[0]
x''''''[0]
-x[t]
-1
1
Perhaps this:
newanswer = Eliminate[{answer, equation}, x''[t]]
(* -x'''[t]] == x[t] *)
Or this:
newanswer = Reduce[{answer, equation}, x'''[t], {x''[t]}]
(* x'''[t]] == -x[t] *)
As for what to do "later on," perhaps this:
Reduce[{answer, equation, x[0] == 1, x'[0] == 2} /. t -> 0]
(* x'''[0] == -1 && x''[0] == 1 && x'[0] == 2 && x[0] == 1 *)
Or this:
Solve[{answer, equation, x[0] == 1, x'[0] == 2} /. t -> 0]
(* {{x[0] -> 1, x'[0] -> 2, x''[0] -> 1, x'''[0] -> -1}} *)