8
$\begingroup$

I would like to know, apart from Method->Automatic what are available methods for DSolve.

sol = DSolve[equationList, f[t], t, Method -> "Holonomic"]

I discovered only "Holonomic". Are there more available methods and if yes for what kind of problems are they good for? I failed to find anything in the documentation.

$\endgroup$
3
  • 3
    $\begingroup$ "EvaluateIntegrals" | "InactiveIntegrals" | "Holonomic" | Automatic are the ones I know.... $\endgroup$
    – Michael E2
    Sep 10, 2022 at 17:26
  • 1
    $\begingroup$ For instance, DSolve[x^2 y''[x] + xy'[x] + y[x] == Exp[4 Log [x]], y[x], x, Method -> "EvaluateIntegrals"] takes 400+ sec. and produces the same answer as DSolve[x^2 y''[x] + xy'[x] + y[x] == Exp[4 Log [x]], y[x], x] in around 0.35 sec. "InactiveIntegrals" seems the same as Automatic, and allows Inactive[] integrals, but does not force them. (Example from mathematica.stackexchange.com/questions/258268/…) $\endgroup$
    – Michael E2
    Sep 10, 2022 at 17:44
  • 2
    $\begingroup$ Found another: eqns = {y'[t] == z[t], z'[t] == -10, y[0] == 1, z[0] == 0}; events = {WhenEvent[y[t] == 0, {y[t] -> 0, z[t] -> -(70/100) z[t]}]}; sol = DSolve[Join[eqns, events], {y[t], z[t]}, {t, 0, 2}, Method -> {"Events", "MaxEvents" -> 3}] $\endgroup$
    – Michael E2
    Sep 10, 2022 at 17:54

1 Answer 1

6
$\begingroup$

Summary

These are what I've discovered (in V13.1):

Method -> Automatic
Method -> "EvaluateIntegrals"
Method -> "InactiveIntegrals" (* same as Automatic? *)
Method -> {"Events", "MaxEvents" -> n}
Method -> "Holonomic"

I don't believe they can be mixed. If you do, what happens varies. Either DSolve returns unevaluated or the combination is treated as Automatic. For instance, Method -> {"EvaluateIntegrals", {"Events", "MaxEvents" -> 3}} is treated as Automatic in "Events" example and leads to unevaluated DSolve[] in the other examples below. Method parsing does not seem to be as robust as in NDSolve or NIntegrate (probably why IncludeSingularSolutions is a separate option and not a Method option).

Method -> "EvaluateIntegrals"

The following takes 400+ sec. and produces the same answer as Method -> Automatic, with unevaluated integrals, in this case:

DSolve[x^2 y''[x] + xy'[x] + y[x] == Exp[4 Log [x]], y[x], x,
 Method -> "EvaluateIntegrals"]

Method -> "InactiveIntegrals"

Method -> "InactiveIntegrals" seems the same as Method -> Automatic, and allows Inactive[] integrals, but does not force them. The following takes around 0.35 sec. with either option setting and produces the same result

DSolve[x^2 y''[x] + xy'[x] + y[x] == Exp[4 Log [x]], y[x], x,
 Method -> "InactiveIntegrals"]

The foregoing examples are adapted from Help!!! DSolve is coming out weirdly

Method -> {"Events", "MaxEvents" -> 3}

The following is adapted from the docs for DSolve and without the "MaxEvents" constraint produces a Piecewise solution with five cases.

eqns = {y'[t] == z[t], z'[t] == -10, y[0] == 1, z[0] == 0};
events = {WhenEvent[y[t] == 0,
 {y[t] -> 0, z[t] -> -(70/100) z[t]}]};
sol = DSolve[Join[eqns, events], {y[t], z[t]}, {t, 0, 2},
 Method -> {"Events", "MaxEvents" -> 3}]

DSolve::maxev: The maximum number of events has been reached. The currently computed solution has been returned.

(* output: Piecewise with three cases *)

Method -> "Holonomic"

For linear equations, it produces a DifferentialRoot:

DSolve[y'[x] == y[x], y, x, Method -> "Holonomic"]
(*
{{y -> DifferentialRoot[
 Function[{\[FormalY], \[FormalX]},
  {-\[FormalY][\[FormalX]] + \[FormalY]'[\[FormalX]] == 0,
  \[FormalY][0] == C[1]}]]}}
*)

Unceremoniously fails on nonlinear equations:

DSolve[y'[x] == y[x]^2, y, x, Method -> "Holonomic"]
(*
DSolve[Derivative[1][y][x] == y[x]^2, y, x, Method -> "Holonomic"]
*)
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.