Answer Happy

Famous Numbers Revisited, One Last Time The following problems describe the numbers e≈ 2.71828182846, In 2≈ 0.69314718056, and ~ 3.14159265359 as specific values of certain initial value problems. In each case, apply the Runge- Kutta method with n = 10, 20, 40, ... subintervals (doubling n each time). How many subintervals are needed to obtain-twice in succession-the correct value of the target number rounded to nine decimal places? 1. The number e = y(1), where y(x) is the solution of the initial value problem dy/dx = y, y(0) = 1. 2. The number In 2 = y(2), where y(x) is the solution of the initial value problem 1/x, y(1) = 0. dy/dx = 3. The number = y(1), where y(x) is the solution of the initial value problem dy/dx = 4/(1+x²), y(0) = 0.