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I would like to resolve symbolically the following equation:

DSolve[{c1[x, y] + c2[x, y] == 0, 
  D[c1[x, y], {x, 1}] == a*Sin[Pi*(y - 1/2)]*Cos[Pi*(y - 1/2)]^3, 
  D[c2[x, y], {y, 1}] == 
  b*Sin[Pi*(x - 1/2)]*Cos[Pi*(x - 1/2)]^3}, {c1[x, y], c2[x, y]}, {x,
  y}]

which should correspond to a system:

$$ \begin{align} c_1\left(x,y\right)+c_2\left(x,y\right) & = 0\\ \frac{\partial c_2}{\partial y} & = a\sin x\cdot\left(\cos x\right)^3 ,\\ \frac{\partial c_1}{\partial x} & = b\sin y\cdot\left(\cos y\right)^3 \end{align} $$

but Mathematica gives me an error:

DSolve::overdet: There are fewer dependent variables than equations, so the system is overdetermined.

What is it so?

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You can solve the differential equations first, then impose the algebraic condition. –  b.gatessucks Apr 21 '13 at 12:41
1  
Why don't you just replace $c2$ with $c1$? DSolve don't handle overdetermined systems of DE even though trivial ones like this one. The reason for that is that makes a simple comparing: NumberofEquation==NumberofDependentVars. –  Spawn1701D Apr 21 '13 at 12:41
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