Lecture 16: Derivation of Implicit Runge-Kutta methods
Now from (6.2) with , we have
,
Expanding by Taylor series about we get
(6.5)
Since these two equations are implicit, we can no longer proceed by successive substitution as done in the case of explicit Runge-Kutta methods earlier. Let us assume instead, that the solutions for and may be expressed in the form
, we have 
, 
by Taylor series about 
we get 
and 
may be expressed in the form 
by (6.6) in (6.5) we get 
