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## Homework Statement

find the solution to:

[itex]\frac{\partial^{2}u}{\partial x \partial y} = 0[/itex]

[itex]\frac{\partial^{2}u}{\partial x^{2}} = 0[/itex]

[itex]\frac{\partial^{2}u}{\partial y^{2}} = 0[/itex]

## Homework Equations

theorem of integration

## The Attempt at a Solution

now from a previous question I had earlier, I have found that I can simply do integration as per normal. So in doing that I managed to get:

[itex]u(x,y) = xf(y) + g(y)[/itex]

[itex]u(x,y) = yf(x) + g(x)[/itex]

However I have a problem that arises when I take the integral of two different variables (in the care of the first expression for u)

[itex]u(x,y) = F(y) + g(x)[/itex] where [itex]F(x) [/itex] is the integral of [itex]f(x)[/itex]

However the final solution is:

[itex]u(x,y) = Ax + By + C[/itex]

...to which I don't how to get to. I understand that when you put back all the partial differentials together, all those arbitrary functions collapse down to one constant of integration. However, I don't see how to get those constant co-efficients in front of x and y. Also I don't know how to treat the [itex]F(x) [/itex] (what to do with it).