Solving the Constant PDE ∂u/∂x=∂u/∂y

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SUMMARY

The discussion focuses on solving the constant partial differential equation (PDE) ∂u/∂x = ∂u/∂y. The established conclusion is that the general solution is u(x,y) = f(x+y), where f is a differentiable function. This indicates that u is not necessarily constant but can vary based on the function f. An example provided is u = 2(x+y), which satisfies the PDE.

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∂u/∂x=∂u/∂y, can we ensure that u is a constant not dependent on x and y?
 
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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"?
 
AlonsoMcLaren said:
∂u/∂x=∂u/∂y, can we ensure that u is a constant not dependent on x and y?

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.
 

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