= Solution
For $y'=f(t,y)$ and step $h$, a <collocation Runge-Kutta method> constructs a <polynomial> $P$ of degree at most $s$ on $[t_n,t_n+h]$ satisfying
$$
P(t_n)=y_n,\qquad
P'(t_n+c_i h)=f(t_n+c_i h,P(t_n+c_i h)),\quad 1\le i\le s.
$$
The step value is $y_{n+1}=P(t_n+h)$. Thus the differential equation holds exactly at the prescribed collocation nodes, while the initial value fixes the integration constant. For a smooth vector field and small enough $h$, the implicit stage equations have the local solution branch continuing the zero-step initial state; arbitrary large steps need not have a unique nonlinear solution.
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