 Solution

Let $e$ be the vector of ones. For a general nonautonomous Runge-Kutta method the stages satisfy $k_i=f(t+c_{i}h,y+h{\sum }_j{a}_{ij}{k}_j)$ and the update is $Y=y+h{\sum }_i{b}_i{k}_i$. Expansion at $(t,y)$ givesand thereforeThe exact solution is $y(t+h)=y+hf+h^2(f_t+f_{y}f)/2+O(h^3)$. Matching the independent terms gives the necessary and sufficient second-order conditions with independent Runge-Kutta abscissaeUnder the usual internally consistent Butcher tableau convention $c=Ae$, this reduces to $c=Ae$, $b^{T}e=1$, $b^{T}c=1/2$. If the abscissae are allowed to be independent of the row sums, $c=Ae$ is a sufficient simplifying convention rather than a logically necessary order-two condition.