How to solve this Laplacian equation?

Question

assume that $C$ is a function of $x$ and $y$,and $\nabla^2 C=0$,we have some specified points of this function($C(0,0)=1 , C(0,1)=2.8 , C(0,2)=5.2 ,C(1,0)=2, C(1,1)=3.8$.I know it's a Laplacian equation but there is not the usual boundary conditions,instead we have some points of our function ,I tried obtaining the general answer for this Laplacian equation(the same answer for Laplacian equations with boundary condition),then I substituted the given points for the variables in the obtained equation but it didn't lead to any useful result(I mean the unknown coefficients wouldn't be discovered this way),any help would be appreciated.

Answer 1

Without boundary conditions, the solution likely won't be unique. If you want just one particular solution, try

$$ f(x,y) = ax^2 + by^2 + cx + dy + e $$

Note that you'll need at least 5 constants to satisfy 5 points, which is why your proposed solution of $(ax + b)(cy + d)$ (in the comments) doesn't work.