A note on trapezoidal methods for the solution of initial value problems

The trapezoidal rule for the numerical integration of first-order ordinary differential equations is shown to possess, for a certain type of problem, an undesirable property. The removal of this difficulty is shown to be straightforward, resulting in a modified trapezoidal rule. Whilst this latent difficulty is slight (and probably rare in practice), the fact that the proposed modification involves negligible additional programming effort would suggest that it is worthwhile. A corresponding modification for the trapezoidal rule for the Goursat problem is also included.