Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 63 / 4 / 1 / Solution

For $y^{\prime }=f(t,y)$ and step $h$, a collocation Runge-Kutta method constructs a polynomial $P$ of degree at most $s$ on $[t_n,t_n+h]$ satisfyingThe 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.