The results of the last problem show that the solutions can be extremely sensitive to changes in the initial conditions. This sensitivity allows chaos to occur in deterministic systems, which is the subject of much current research.
One way to experience first hand the sensitivity to changes in the initial conditions is to try a little “target practice.” For the ODEs in Exercise, use dfield6 to find approximately the value of xo such that the solution x(t) to the initial value problem with initial condition x(0) = x0 satisfies x(t1) = x1. You should use the Keyboard input window to initiate the solution. Widen the window to allow a large number of digits in the edit window by clicking and dragging on the right edge. After an unsuccessful attempt try again with another initial condition. The Uniqueness Theorem should help you limit your choices. If you make sure that the Display Window is the current figure (by clicking on it), and execute plot (t1, x1, ‘or’) at the command line, you will have a nice target to shoot at.
You will find that hitting the target gets more difficult in each of these problems. We allow you to “cheat” by starting a solution in the target, and finding the value at t = 0. However, be sure to try to hit the target with that initial value. You may be surprised at the outcome.
x′ = x(l − x2), t1 = 5, x1 = 0.5.
We need at least 10 more requests to produce the solution.
0 / 10 have requested this problem solution
The more requests, the faster the answer.