CSC 210

Week 6



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

  2. An explosion problem: Here we consider the "explosion" problem x'= x^2, x(0)=1. This simple equation shows up in chemical reactions where two atoms get together to form a molecule.

  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. Euler for systems of ODEs: Euler's algorithm can be generalized for systems of ODEs. For example, for the pair of differential equations

    dx/dt = f (x, y)
    dy/dt = g (x, y)

    with initial conditions x(0) = x_0 and y(0) = y_0, Euler's algorithm with step size h becomes

    x_(n+1) = x_n + h*f(x_n, y_n)
    y_(n+1) = y_n + h*g(x_n, y_n).

    Now consider the Epidemics problem from last week. Take a = 0.6, r = 0.003, S_0 = 200, I_0 = 20, and step size h = 0.2. Compute two steps of Euler by hand. Compare your numbers with those from Phaser.

  5. Reading Euler: Try to read the original paper of Euler listed above. Is his algorithm the same as the one we derived in class? What does he have to say about errors? This problem is for your own edification; you do not have to turn it in.