I want to find the unknown constants in the function $f(x,y)=A(e^{-i.k_{x}x}+C_{1}x+C_{2})(e^{-i.k_{y}y}+C_{3}y+C_{4})$, using the following known boundary conditions and auxiliary equation (contraint), where $A, C_{1}, C_{2}, C_{3}, C_{4}$ are the wanted unknown constants (coefficients) and $(x,y)$ are the independent variables.
The conditions are: $(\partial f/\partial y)_{y=0}=(\partial f/\partial y)_{y=y_{0}}=(\partial f/\partial x)_{x=0}=(\partial f/\partial x)_{x=x_{0}}=0$
The auxiliary equation (contraint) is: $\int\limits^{u_{2}}_{u_{1}}\int\limits^{v_{2}}_{v_{1}}f((u-v)/\sqrt{2},(u+v)/\sqrt{2})du dv$. Note that this is simply integrating $f(x,y)$ but after changing variables to: $x=(u-v)/\sqrt{2}$ and $y=(x+v)/\sqrt{2}$.
Note that $x_{0},y_{0},u_{1},u_{2},v_{1},v_{2}$ are all known constants.
The number of unknown constants ($A, C_{1}, C_{2}, C_{3}, C_{4}$) in the function $f$ is 5. The number of expressions given above (1 equation and 4 boundary conditions) is also 5. So, in principle, we should be able to find these constants (coefficients).
How do I determine these constants from the information given above? One way I thought of is to re-arrange the resulting 5 expressions in matrix form, ordered by these coefficients, then enforce a nontrivial solution by taking the determinant = zero. However, two questions arise here:
1- How to re-write these expressions in terms of these coefficients (constants)? Especially if some terms have more than one (mixed) coefficients, like say $C_{1}C_{3}$ or $AC_{3}$?
2- Is there an easy way to make this using a computer tool (Mathematica or MATLAB)? If so, please indicate how?
I am also open to other suggestions on how to solve this.
Thanks.