DSolve not evaluating initial condition

Question

I have a system of 1st order equations (it's overdetermined but well posed) that is solved within an arbitrary constant:
f[u_, v_] := Cos[u - v]; 
DSolve[{D[x[u, v], u] == Cos[u - v] f[u, v], D[x[u, v], v] == Sin[u - v] D[f[u, v], v]}, x[u, v], {u, v}]

Fine, I get a solution. Now, when I impose the initial condition $x(0,0)=0$ Mathematica does not return anything
DSolve[{D[x[u, v], u] == Cos[u - v] f[u, v], D[x[u, v], v] == Sin[u - v] D[f[u, v], v], x[0, 0] == 0},  x[u, v], {u, v}]

but I get just the same code I introduced.
I can take the initial solution with the arbitrary constant and just solve a linear equation, but I would like to understand why the initial condition is not automatically evaluated by DSolve.

Spawn1701D · Answer

For your kind of problem, instead of solving the compatible system you can just solve one of them with a condition of the type $x(0,v)=0$. For instance
DSolve[{D[x[u, v], u] == Cos[u - v] f[u, v], u[0,v]==0}, x[u, v], {u, v}]

Now, why DSolve doesn't solve the problem how you posed it? It expects different kind of conditions, see DSolve's help for more info. In general defining only a point means nothing for a PDE because its solution will depend on a number of arbitrary functions. In your case the solution depends on a constant that's why $x(0,0)=0$ works.

DSolve not evaluating initial condition

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