# DSolve not evaluating initial condition

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.

• With Maple 2021 I have: $$x(u,v)=\frac{1}{4} \sin (2 u-2 v)+\frac{u}{2}+\frac{v}{2}$$ Jul 24 at 9:21
• Why do you call it an initial condition? It is a boundary condition---and this is the problem---you do not define it at one point. Jul 26 at 12:42
• @yarchik, Such systems of equations have unique solutions up to a constant, which means that we fix the constant at a unique point, not along a curve. Regardless the name that you may give to such conditions, I do not see why DSolve should not evaluate it. Jul 26 at 14:22
• Did you report it to WRI? They are the ones that improve the code. (Or maybe they'll tell you how to properly set it up. But I'm with you: This should, or they should make it, workjust as you set it up.) Aug 29 at 23:32
• A similar problem in the documentation: DSolve[{D[f[x, y], x] == x y Cos[x y] + Sin[x y], D[f[x, y], y] == -E^-y + x^2 Cos[x y]}, f, {x, y}]. Recover a function from its gradient vector. The solution represents a family of parallel surfaces. Aug 30 at 0:53

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.

According to the documentation:

(*Recover a function from its gradient vector*)
sol = DSolve[{D[x[u, v], u] == Cos[u - v] Cos[u - v],
D[x[u, v], v] == Sin[u - v] D[Cos[u - v], v]}, x, {u, v}][]

(*{x -> Function[{u, v}, v + C + 1/4 (2 u - 2 v + Sin[2 (u - v)])]}*)


The solution represents a family of parallel surfaces:

Plot3D[Table[x[u,v]/.sol[]/.{C->k},{k,1,16,4}]//Evaluate,
{u,0,15},{v,0,2},PlotRange->All] • True, it's exactly the same. It's not an initial or boundary problem (actually these names are essentially defined for second and higher order of derivatives), but we just need a point in space to have a unique solution. Aug 30 at 7:11