CSC 210

Week 6



  1. Largest safe step size: Consider the differential equation x'= -12x.

  2. Impossible computation: Consider the "explosion" problem x'= x^2, x(0)=1, and try to compute x(1) using several different algorithms with various step sizes.

  3. Two steps of Improved Euler(2): Consider the initial value problem x1' = x, x(0)=1. With step size h = 0.5, compute, by hand using paper and pencil, two steps of Improved Euler (2) algorithm to obtain the approximate value of x(1). You can look up the formula for Improved Euler (2) in Phaser Help; make sure you show all the intermediate numbers Ks. Now compare your answer with the one you get from Phaser.

  4. Nonautonomous Euler: When the right-hand-side of a differential equation contains time t explicitly, the equation is called nonautonomous. Now, consider a differential equation of the form dx/dt = f (t, x) with x(t_0) = x_0. Then Euler becomes
    x_(n+1) = x_n + h*f(x_n, t_n)
    t_(n+1) = t_n + h .
    Now, in the Gompertz equation x' = a*(exp(-b*t))*x, set a = 3 and b = 2.1, and x(0) = 4.5; and take step size h = 0.2. By hand (with the help of a calculator) compute two steps of Nonutonomous Euler on Gompertz and compare your numbers to the ones you get from PHASER.

  5. Reading Euler: Try to read the original paper of Euler listed above. Write a short commentary on this paper. Is his algorithm the same as the one we derived in class? What does he have to say about errors?