Euler’s Midpoint Method is a variation on Euler’s Method that is significantly more accurate in general. For time step h and initial value y₀ = y(t₀), the values yk are defined successively by

Apply both Euler’s Method and the Euler Midpoint Method with h = 0.1 to estimate y(1.5), where y(t) satisfies dy/dt = y with y(0) = 1. Find y(t) exactly and compute the errors in these two approximations.

Yk = yk-1 + hmk-1 where mk-1 = F (_- + / 0- + / F(~392-1)). h h 1 2.3k-1