Take some simple function of 2 variables, e.g. f[x_, y_]:= x+y and try to solve a PDE of it for the general solution:
DSolve[{Derivative[1, 0][f][x, y] == 1, Derivative[0, 1][f][x, y] == 1}, f, {x, y}]
This works as expected. However, if you try to give an initial condition for a particular solution, Mathematica is no more able to solve it:
DSolve[{Derivative[1, 0][f][x, y] == 1,
Derivative[0, 1][f][x, y] == 1, f[0, 0] == 0}, f, {x, y}]
Has anyone an explanation?
(previous comment removed) I’d like to show a step proof of the 2nd DSolve result or say nothing - I do not have it at the moment.
Thank’s for your answer. I have some questions.
Why is the initial condition f[0,0] = 0 impossible for f[x_,y_]=x+y?
What do you mean by: " choice of f[x,y] which you did not give". DSolve is supposed to give f[x,y], not I.
You seem to imply that Mathematica solves the equations in sequence. Well let’s do this. First the first diff.equ. then the second, then the initial condition :
f1 = f /. DSolve[D[f[x, y], x] == 1, f, {x, y}][[1]]
f2 = f1 /. DSolve[{D[f1[x, y], y] == 1}, C[1], y][[1]]
f3 = f2 /. Solve[f2[0, 0] == 0, C[2]][[1]]
Seems to work, f3 is the solution.
Why do you think: “With coupling (you gave a point of intersection of), you cannot express a function that will work for all x and all y for both equations simultaneously.”. Well {0,0} is such a point of intersection that works for all x and y.
I had made a careless error in my head while was assuming mm was right. I have no proof to show you at this time why the internals of Mathematica made it’s decision and would rather say nothing as I said before.
But since you re-asked, I will say this is not an infrequent question on Communities and that often questioners have set up conditions incorrectly. In this case I will say “valid conditions do not always help Mathematica solve a PDE”. The code used to solve is on your system (you can find it through Help). But that is not an answer because I have not read “all of how mm solves each PDE” I cannot answer you directly, as to internals. I wish to avoid taking an unresolved question and adding further guesses.
To err is human, but it is great if someone stands to it. Thank’s.
The Help documentation mentions choosing c1 after solving (not during), I’m unsure if there’s a reason why. This may hint that, in effect, one should select a function rather than a constant. However, this is guess work.
DSolveValue[{D[f[x, y], x] + D[f[x, y], y] == 2}, f[x, y], {x, y}] == 2 x + C[1][-x + y]
DSolveValue[{D[f[x, y], x] - D[f[x, y], y] == 0}, f[x, y], {x, y}] == C[1][x + y]
DSolveValue[{D[f[x, y], x] + D[f[x, y], y] == 2, f[x, 0] == x},
f[x, y], {x, y}] == x+y
But that is a poor answer, since combining these still does not solve, even though Mathematica “knows how to solve systems of PDE”, and because it is a guess not a fact, and because the above is not your original system. The problem is then to know how the internals work exactly, and I have not memorized all of the internals.
DSolveValue[{D[f[x, y], x] - D[f[x, y], y] == 0,
D[f[x, y], x] + D[f[x, y], y] == 2, f[x, 0] == x, f[0, y] == y},
f[x, y], {x, y}] == Identity