Second-order conditions with independent Runge-Kutta abscissae ($b^{T}e=1,b^{T}c=b^{T}Ae=1/2$)

For a general nonautonomous Runge-Kutta method, Taylor expansion of stage $i$ gives $k_i=f+h[c_i{f}_t+(Ae{)}_i{f}_{y}f]+O(h^2)$. Matching the exact increment $hf+h^2(f_t+f_{y}f)/2$ gives the three displayed conditions. When internal consistency $c=Ae$ is imposed, the last two coincide. Internal consistency is a usual tableau convention, but is not logically necessary for order two of a generalized tableau.