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I am trying to solve a complicated system of differential equations which can be reduced to a MWE of this kind:

NDSolve[{c'[x] == 10 d[x, y] + D[d[x, y], x], D[d[x, y], {x,2}] + D[d[x, y], {y,2}] == 1, c[0] == 0, DirichletCondition[d[x, y] == 0, True]}, {c, d}, {x, 0,   1}, {y, 0, 1}]

There is one function $c$ that only depends on a variable $x$, while the other function $d$ depends on both $x,y$. Trying to execute that code I get the error

"There are fewer dependent variables, {d[x,y]}, than equations, so \ the system is overdetermined."

How can I solve it?

I use version 11.3 for Linux if needed

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  • $\begingroup$ Can you treat c[] formally as a function of x and y, even though as a function of y it is constant? $\endgroup$
    – Michael E2
    Oct 19 '20 at 13:48
  • $\begingroup$ @MichaelE2 I think it is possible $\endgroup$
    – mattiav27
    Oct 19 '20 at 13:49
  • $\begingroup$ @cvgmt d[x, y] must satisfy {D[d[x, y], {x, 2}] + D[d[x, y], {y, 2}] == 1, D[10 d[x, y] + D[d[x, y], x], y] == 0}. $\endgroup$
    – bbgodfrey
    Oct 19 '20 at 20:02
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Maybe split the question into two differential equations.

Update

Clear["`*"];
sol = NDSolve[{D[d[x, y], {x, 2}] + D[d[x, y], {y, 2}] == 1, 
   DirichletCondition[d[x, y] == 0, True]}, d, {x, 0, 1}, {y, 0, 1}]
NDSolve[{D[c[x, y], x] == 10 d[x, y] + D[d[x, y], x], 
   c[0, y] == 0} /. sol, c, {x, 0, 1}, {y, 0, 1}]

Original

Clear["`*"];
sol = NDSolve[{D[d[x, y], {x, 2}] + D[d[x, y], {y, 2}] == 1, 
   DirichletCondition[d[x, y] == 0, True]}, d, {x, 0, 1}, {y, 0, 1}]
With[{y = .1}, 
 NDSolve[{c'[x] == 10 d[x, y] + D[d[x, y], x], c[0] == 0} /. sol, 
  c, {x, 0, 1}]]
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    $\begingroup$ Solution c[x,y] should fullfill Derivative[0,1][c][x,y]==0 $\endgroup$ Oct 19 '20 at 17:24
  • $\begingroup$ @UlrichNeumann How do I add that condition? $\endgroup$
    – mattiav27
    Oct 20 '20 at 6:43
  • $\begingroup$ @mattiav27 Actually I don't know. My intention was to show that the solution given by cvgmt isn't the one you are looking for. $\endgroup$ Oct 20 '20 at 6:52

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