Equivalent formulations of Runge Kutta Methods? I have given two formulations of a general Runge Kutta Method with $s$ stages


*

*$$ y(x+h) = y_0 + h\sum_{i=1}^{s}b_ik_i$$
$$ k_i = f\left(x_0+c_ih;\,y_0+h\sum_{i=1}^{s}a_{ij}k_j\right)$$

*$$ y(x+h) = y_0 + h\sum_{i=1}^{s}b_if\left(x_0+c_ih;\,y_i\right) $$
$$ y_i = y_0 + \sum_{i=1}^{s}a_{ij}f\left(x_o+c_ih;\, y_i\right)$$


Those forumalas produce the same approximation, do they? But how can one show this?
Just putting in the definition of $k_i$ or respectivly $y_i$ leads to an "endless cycle".

Set $y(x+h)$ from the first definition equal to $y(x+h)$ from the second, and show that they result in the same thing (which they should, because the definitions are the same).
