How to solve a system of differential equation like this one

Question

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

Answer 1

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}]]