CSC 210

Week 6


Topics:


Assignment:

  1. A culture of 8 bacteria are placed in a test tube, and their numbers is counted daily. At first, when their numbers was small they grew at a rate of %180 per day. After a few days, their numbers stabilized at about 405.

  2. Consider the differential equation x'= -10x. Show, using the Linearization Theorem, that x=0 is an asymptotically stable equilibrium point. Now, in the XiVsTime View of Phaser, compute several solutions starting near the equilibrium point using Euler's algorithm with various step sizes. Use a large graph point size and connect points. Determine the largest step size that results in a picture where the origin looks like an asmyptotically stable point. Hint: h = 0.3 does not give a correct picture.

  3. Use PHASER for solving the initial-value problem x' = x, x(0) = 1 at t=1, that is x(1)= e = 2.7182818284590452354 with several algorithms as follows. You should open the Console in the Numerics Editor to get the stats for your computations:

  4. 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.

  5. 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(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, and x(0) = 5; and take step size h = 0.1. By hand compute two steps of Nonutonomous Euler on Gompertz and compare your numbers to the ones you get from PHASER.