**MSC:**- 65L05 Initial value problems

Runge--Kutta methods for ordinary differential equations to

half-explicit methods for differential-algebraic systems of

index 2 results in methods of order {$q\leq 2$}.

The construction of higher order methods is simplified

substantially by a slight modification of the method

combined with an improved strategy for the computation of

the algebraic solution components. We give order conditions

up to order {$q=5$} and study the convergence of these methods.

Based on the fifth order method of Dormand and Prince a

half-explicit Runge--Kutta method of order {$q=5$} is

constructed that requires the solution of 6 systems of

nonlinear equations per step of integration.