∂u/∂x=∂u/∂y, can we ensure that u is a constant not dependent on x and y?
It says that the gradient of u in the x direction is the same as in the y direction for all (x,y).
What sort of function has that property?
What do you mean by "ensure"?
The general solution is u(x,y)=f(x+y), where f is a differentiable function.
So u can be dependent on x and y.
For instance u=2(x+y) is a solution.
Separate names with a comma.