Modified Euler Method

The objective in numerical methods is, as always, to achieve the most accurate (and reliable!) result with the least effort. For integrating the initial value problem (3) the effort is usually measured by the number of times the function $f(t,y)$ must be evaluated in stepping from $a$ to $b$. As we will see, a simple improvement doubles the number of function evaluations per step, but yields a second order method - a winning strategy.