How to solve a system of partial differential equations?

Question

Edit: since the upgrade to Mathematica 10, this problem seems solved

I just want to solve a system of partial differential equations, for example:

$$ \left\{ \begin{array}{l} \frac{\partial}{\partial a}[f(a, b, c)] = 4 \sin^2(b) \cos(c) \\ \frac{1}{a} \times \frac{\partial}{\partial b}[f(a, b, c)] = 4 \cos(c) \sin(2b) \\ \frac{1}{a \sin(b)} \times \frac{\partial}{\partial c}[f(a, b, c)] = -4 \sin(b) \sin(c) \\ \end{array} \right. $$

And when I try to solve this system in Mathematica, the output does not help:

    DSolve[
    {
        D[f[a, b, c], a] == 4 Sin[b]^2 Cos[c],
        (1/a) *D[f[a, b, c], b] == 4 Cos[c] Sin[2 b],
        (1/(a Sin[b]))*D[f[a, b, c], c] == -4 Sin[b] Sin[c]
    }, f[a, b, c], {a, b, c}]


(* DSolve[
    {
        Derivative[1, 0, 0][f][a, b, c] == 4*Cos[c]*Sin[b]^2,
        Derivative[0, 1, 0][f][a, b, c]/a == 4*Cos[c]*Sin[2*b],
        (Csc[b]*Derivative[0, 0, 1][f][a, b, c])/a == -4*Sin[b]*Sin[c]
    }, f[a, b, c], {a, b, c}] *)

Obviously, this output is useless… Probably I doing something wrong…

Thank you for your help

Note : The solution is $f(a, b, c) = 4a \sin^2(b) \cos(c) + K$ (K : the integration constant).

Answer 1

One can get a hint of the issue by seeing that

 DSolve[{1/a D[f[a, b], a] == 1, D[f[a, b], b] == 1}, f[a, b], {a, b}]

can't be solved, but

 DSolve[{ D[f[a, b], a] == a, D[f[a, b], b] == 1}, f[a, b], {a, b}]
 (* {f[a, b] -> a^2/2 + b + C[1]}} *)

can. But there are the same system! (multiplying by a both sides of the first equation in the first case gives the second system). The first system, as written, is consistent. There is a test one can use to check the system of PDE's is consistent. Using Maple:

with(PDEtools):

eq1 := 1/a*diff(f(a,b),a) = 1:
eq2 := diff(f(a,b),b)     = 1:

ConsistencyTest({eq1,eq2});
(* true *)
dsolve({eq1,eq2},f(a,b));
(* f(a, b) = (1/2)*a^2+b+_C1 *)

May be DSolve got worried about singularity when a=0? I do not know. But noticing the above gives a hint on the solution. Simply rearrange terms so that the leading derivative term has unity as factor.

DSolve[{
  D[f[a, b, c], a] == 4 Sin[b]^2 Cos[c],
  D[f[a, b, c], b] == a 4 Cos[c] Sin[2 b],
  D[f[a, b, c], c] == -a Sin[b] 4 Sin[b] Sin[c]},
 f[a, b, c], {a, b, c}
 ]

(* {{f[a, b, c] -> C[1] + 4 a Cos[c] Sin[b]^2}} *)

By the way, there really should not be a need to do this rearrangement. Maple 17 can solve this as is

The real question is: Why is the rearrangement needed? that is what the inquiring minds want to know :)

Answer 2

In Mathematica 10, one can use the original DSolve entry by embedding it within an Assuming

Assuming[a != 0, DSolve[. . .] ]