I have a system of equations:
\begin{align} & x_{21} (\frac{\partial}{\partial x_{11}}f_{1111})( x_{11} , x_{21}, y_{11} , y_{21} ) + \frac{y_{21}}{x_{11}^2} (\frac{\partial}{\partial y_{11}}f_{1111})( x_{11} , x_{21}, y_{11} , y_{21} ) \\ & = f_{2111}(x_{11}, x_{21}, y_{11}, y_{21}) + x_{11}^{-1}x_{22} f_{1121}( x_{11}, x_{21}, y_{11}, y_{21} ) - \frac{1}{2} x_{22} y_{21}. \end{align}
\begin{align} & x_{21} (\frac{\partial}{\partial x_{11}}f_{1121})( x_{11} , x_{21}, y_{11} , y_{21} ) + \frac{y_{21}}{x_{11}^2} (\frac{\partial}{\partial y_{11}}f_{1121})( x_{11} , x_{21}, y_{11} , y_{21} ) \\ & = f_{2121}(x_{11}, x_{21}, y_{11}, y_{21}). \end{align}
\begin{align} & x_{21} (\frac{\partial}{\partial x_{11}}f_{ 2121})( x_{11} , x_{21}, y_{11} , y_{21} ) + \frac{y_{21}}{x_{11}^2} (\frac{\partial}{\partial y_{11}}f_{2121})( x_{11} , x_{21}, y_{11} , y_{21} ) = 0. \end{align}
\begin{align} & x_{21} (\frac{\partial}{\partial x_{11}}f_{ 2111})( x_{11} , x_{21}, y_{11} , y_{21} ) + \frac{y_{21}}{x_{11}^2} (\frac{\partial}{\partial y_{11}}f_{2111})( x_{11} , x_{21}, y_{11} , y_{21} ) \\ & = \frac{1}{2x_{11}} x_{21} x_{22} y_{21} + \frac{1}{x_{11}} x_{22} f_{2121}(x_{11}, x_{21}, y_{11}, y_{21}). \end{align}
\begin{align} \frac{1}{4} x_{11} y_{21} = f_{1121}(x_{11}, x_{21}, y_{11}, y_{21}). \end{align}
Here $f_{1111}, f_{2111}, f_{1121}, f_{2121}$ are functions in $x_{11}, x_{21}, y_{11}, y_{21}$.
Are there some software which can solve this system of equations exactly? Thank you very much.
One solution of this system of equations is \begin{align} f_{1111} & = - \frac{1}{4}x_{11} y_{11}, \\ f_{1121} & = \frac{1}{4} x_{11} y_{21}, \\ f_{2111} & = - \frac{1}{4} x_{21} y_{11}, \\ f_{2121} & = \frac{1}{4} x_{21} y_{21}. \end{align}