On those ordinary differential equations that are solved exactly by the improved Euler method

summary:As a numerical method for solving ordinary differential equations $y^{\prime }=f(x,y)$, the improved Euler method is not assumed to give exact solutions. In this paper we classify all cases where this method gives the exact solution for all initial conditions. We reduce an infinite system of partial differential equations for $f(x,y)$ to a finite system that is sufficient and necessary for the improved Euler method to give the exact solution. The improved Euler method is the simplest explicit second order Runge-Kutta method