open Expr
open Simplify
open Integrate

type ode = {
  equation: expr;
  dependent: string;
  independent: string;
}

let dsolve ode_eq dep_var indep_var =
  let try_separation_of_variables () =
    match ode_eq with
    | Mul (f_y, g_x) ->
        (match (integrate indep_var g_x, integrate dep_var f_y) with
         | Some int_g, Some int_f ->
             Some (simplify (Sub (int_f, int_g)))
         | _ -> None)
    | _ -> None
  in

  let try_linear_first_order () =
    None
  in

  let try_exact_equation () =
    None
  in

  let try_integrating_factor () =
    None
  in

  match try_separation_of_variables () with
  | Some sol -> Some sol
  | None ->
      match try_linear_first_order () with
      | Some sol -> Some sol
      | None ->
          match try_exact_equation () with
          | Some sol -> Some sol
          | None -> try_integrating_factor ()

let solve_second_order coeff_y'' coeff_y' coeff_y rhs var =
  let a = coeff_y'' in
  let b = coeff_y' in
  let c = coeff_y in

  match (a, b, c) with
  | (Const a_val, Const b_val, Const c_val) when rhs = Const 0.0 ->
      let discriminant = b_val *. b_val -. 4.0 *. a_val *. c_val in
      if discriminant > 0.0 then
        let r1 = (-.b_val +. sqrt discriminant) /. (2.0 *. a_val) in
        let r2 = (-.b_val -. sqrt discriminant) /. (2.0 *. a_val) in
        Some (Add (
          Mul (Var "C1", Exp (Mul (Const r1, Var var))),
          Mul (Var "C2", Exp (Mul (Const r2, Var var)))
        ))
      else if discriminant = 0.0 then
        let r = -.b_val /. (2.0 *. a_val) in
        Some (Mul (
          Add (Var "C1", Mul (Var "C2", Var var)),
          Exp (Mul (Const r, Var var))
        ))
      else
        let real_part = -.b_val /. (2.0 *. a_val) in
        let imag_part = sqrt (-.discriminant) /. (2.0 *. a_val) in
        Some (Mul (
          Exp (Mul (Const real_part, Var var)),
          Add (
            Mul (Var "C1", Cos (Mul (Const imag_part, Var var))),
            Mul (Var "C2", Sin (Mul (Const imag_part, Var var)))
          )
        ))
  | _ -> None

let solve_system _odes _indep_var =
  []