It is given that \(y=m x^{2}+\frac{x}{2}+n\), where \(m\) and \(n\) are non-zero constants. It is also given that \(3\left(\frac{\mathrm{~d}^{2} y}{\mathrm{~d} x^{2}}\right)=\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)^{2}-y\) for all values of \(x\). Find the values of \(m\) and \(n\).