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}}  *)

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