I have the following set of nonlinear equations
$f_{11}(U)A+f_{12}(U)B+f_{13}(U)A^2+f_{14}(U)B^2+f_{15}(U)AB=0$ $f_{21}(U)A+f_{22}(U)B+f_{23}(U)A^2+f_{24}(U)B^2+f_{25}(U)AB=0$
where $f_{ij}$ are functions of the variable $U$.
In my problem, $A$ and $B$ are small, but non-zero.
An approximate solution for this problem can be found by neglecting the higher order terms: $A^2, B^2, AB$ (since $A$ and $B$ are small). This allows to write:
$f_{11}(U)A+f_{12}(U)B=0$
$f_{21}(U)A+f_{22}(U)B=0$
These equations can be written in matrix format
$\left[\begin{array}{cc} f_{11}(U) & f_{12}(U) \\ f_{21}(U) & f_{22}(U) \end{array}\right] \left[\begin{array}{c} A \\ B \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \end{array}\right]$
For non-trivial solutions (non-zero $A$ and $B$), the determinant of this matrix should be zero.
$f_{11}(U) f_{22}(U) - f_{12}(U) f_{21}(U) = 0 $
This can be solved to find the values of $U$ (which is the result of interest in this problem).
My question is how can I solve this problem without neglecting the higher order terms: $A^2, B^2, AB$.