Labelling vector state variables for solution of system of ODEs

Question

This might be a duplicate, but I have not seen this problem addressed for DSolve usage.

I have a system of ODEs of the form

Y' = AY + B

When we put this into Mathematica as a vector variable:

sol = FullSimplify[DSolve[y'[t] == A . y[t], y \[Element] Vectors[n], t]]

We get a pure function solution where each state variable solution is just separated by a comma. I would like each state variable in the solution to be labelled in the standard way:

vec_{y} = {y_{1}, y_{2}, ..., y_{n}} = {y_{1} = c1*e^(-10t)+..., ...}

So that I can also subset the solution depending on which variable I am interested in, for example sol[[y1]] or sol[[1]].

Answer 1

I'm guessing the output of a working DSolve call is something like this:

sol = {{y -> Function[t, Evaluate@Table[Exp[n t], {n, 3}]]}}
(* {{y -> Function[t, {E^t, E^(2 t), E^(3 t)}]}} *)

Then this splits it up and indexes the solution components. It's not clear to me how you wanted the components constructed. Indexed symbols are the usual way to code subscripts, but the y[k] can be changed to whatever suits you.

{{y -> Function[t, Evaluate@Table[Exp[n t], {n, 3}]]}} /. 
   {y_ -> Verbatim[Function][t_, body_List]} :> 
  Table[y[k] -> Function[t, Evaluate[body[[k]]]], {k, Length@body}]
(*
{{y[1] -> Function[t, E^t], y[2] -> Function[t, E^(2 t)], 
  y[3] -> Function[t, E^(3 t)]}}
*)