mproved Euler's Method The improved Euler's method is obtained by using an average of f values, i.e., a slope average: y(t1) y(t0) h f (t0, y(t0)) f (t1, y(t1)) 2 (3) Then substitute y(t0) = y0 and define y1 as this approximate value of y(t1), i.e. y1 = y0 h 2 (f (t0, y0) f (t1, y1)) . (4) The equation (4) defines the trapezoidal method. Unfortunately, this formula defines y1 only implicitly, since y1 appears on both sides of the equality so that an equation must be solved to obtain y1. To avoid this problem, and since we already have made an approximation to get (4), we replace y1 on the right-hand side by the approximation one would obtain by simply applying Euler's method from (t0, y0). The resulting quantity y1 = y0 h 2 ( f (t0, y0) f (t1, y0 hf (t0, y0) Euler y1 from (1) ) ) (5) with t1 = t0 h is the improved Euler approximation. This approximation can be thought of as a correction to the Euler approximation. The iteration (5) is then repeated to obtain y2 y(t2), . . ., i.e., f1 = f (tn, yn)