# Unknown (internal?) function with DSolve

Bug introduced in 8.0 or earlier and fixed in 10.3.0

I am solving With Mathematica 10 the following ODE system:

DSolve[{B'[x] == -f[x]*Cos[l]*G[x], G'[x] == +f[x]*Sin[l]*B[x]}, {B,G}, x]


The solution is almost trivial, but Mathematica gives me this expression:

What is

(* -DSolveDSolveFirstOrderODEsDumpconst\$26192[2]*)


I can't find it in any tutorial online.

Is this a bug? I cannot check with another version...

• It's a bug. Just consider it to be a second constant parameter, C[2]. May 31, 2015 at 17:16
• Same in version 8 May 31, 2015 at 17:28
• @ilian Taking it as a second constant, this solution appears to be correct. But in version 10.2.0, DSolve returns unevaluated instead of returning this solution. Is that a separate, new-in-10.2 bug? Jul 24, 2015 at 19:23
• @Szabolcs I've mentioned this to the developers and they will be looking into it. Jul 24, 2015 at 20:19

As of Mathematica 10.3, a solution is returned by DSolve

eqns = {B'[x] == -f[x]*Cos[l]*G[x], G'[x] == +f[x]*Sin[l]*B[x]};
sol = DSolve[eqns, {B, G}, x]

(* {{B ->
Function[{x},
1/2 (-2 E^-Integrate[-I f[K[1]] Sqrt[Cos[l] Sin[l]], {K[1], 1, x},
Assumptions -> True] + E^
Integrate[-I f[K[1]] Sqrt[Cos[l] Sin[l]], {K[1], 1, x},
Assumptions -> True]) C[1] +
1/2 E^(1 +
Integrate[-I f[K[1]] Sqrt[Cos[l] Sin[l]], {K[1], 1, x},
Assumptions -> True]) C[2]],
G -> Function[{x},
1/2 I E^(
1 + Integrate[-I f[K[1]] Sqrt[Cos[l] Sin[l]], {K[1], 1, x},
Assumptions -> True]) C[2] Sec[l] Sqrt[Cos[l] Sin[l]] - (1/(
2 f[x]))C[1] Sec[
l] (-2 I E^-Integrate[-I f[K[1]] Sqrt[Cos[l] Sin[l]], {K[1], 1,
x}, Assumptions -> True] f[x] Sqrt[Cos[l] Sin[l]] -
I E^Integrate[-I f[K[1]] Sqrt[Cos[l] Sin[l]], {K[1], 1, x},
Assumptions -> True] f[x] Sqrt[Cos[l] Sin[l]])]}} *)

Simplify[eqns /. sol]

(* {{True, True}} *)

• Can you please take a look here? I think it's a bug that with the new DateString specifications only a single tick mark is shown in these examples. I don't want to send support requests about other people's problems, but I wanted to make sure that someone at WRI will notice ... Oct 21, 2015 at 12:23