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  • $\begingroup$ When DSolve returns unevaluated, it means that it couldn't solve the problem. From the documentation: "DSolve can find general solutions for linear and weakly nonlinear partial differential equations. Truly nonlinear partial differential equations usually admit no general solutions." While yours looks solvable, it probably just decides it can't do it. $\endgroup$
    – Szabolcs
    Commented Feb 14, 2014 at 21:46
  • $\begingroup$ That's strange. Because this system has at least one solution : f(a, b, c) = 4a Sin[b]^2 * Cos[c] + K (K : the integration constant). $\endgroup$
    – mlpo
    Commented Feb 14, 2014 at 21:49
  • 2
    $\begingroup$ Here is something more strange. Try DSolve[{D[f[a, b], a] == 0,(1/a) D[f[a, b], b] == 0},f[a, b], {a, b}] and it can't do it. Now remove the (1/a) from the second equation, (which is the same as multiplying both sides by a, then it solves it ! $\endgroup$
    – Nasser
    Commented Feb 14, 2014 at 22:34
  • 1
    $\begingroup$ @Nasser Looks like you touched on the solution. If we bring the OP's PDE system into a form where the LHS only has the derivatives, Mma can solve the system: DSolve[{D[f[a, b, c], a] == 4 Sin[b]^2 Cos[c], D[f[a, b, c], b] == 4 a Cos[c] Sin[2 b], D[f[a, b, c], c] == -4 a Sin[b] Sin[b] Sin[c]}, f[a, b, c], {a, b, c}] You should post this as an answer, it's worth highlighting. $\endgroup$
    – Szabolcs
    Commented Feb 14, 2014 at 22:37
  • 1
    $\begingroup$ @Szabolcs Yes, I was just trying to verify things...was planing to write something but have to make coffee first :) btw, Maple is able to solve it in its current form without rearranging: screen shot !Mathematica graphics $\endgroup$
    – Nasser
    Commented Feb 14, 2014 at 22:39