ABSTRACT
Because many simulations still involve differential equations we begin by considering the simplest first order differential equations of the form
You recall a direction field is simply drawing at each point in the x-y plane a line element with the slope given by the differential equation, Figure 20.I. For example, the differential equation
has the indicated direction field, Figure 20.II. On each of the concentric circles
the slope is always the same, the slope depending on the value of k. These are called isoclines. Looking at the following picture, Figure 20.III, the direction field of another differential equation, on the
left you see a diverging direction field, and this means small changes in the initial starting values, or small errors in the computing, will soon produce large differences in the values in the middle of the trajectory. But on the right hand side the direction field is converging, meaning large differences in the middle will lead to small differences on the right end. In this single example you see both small errors can become large ones, and large ones can become small ones, and furthermore, small errors can become large and then again
become small. Hence the accuracy of the solution depends on where you are talking about it, not any absolute accuracy over all. The function behind all this is
whose differential equation is, upon differentiating,
Probably in your mind, you have drawn a “tube” about the “true, exact solution” of the equation, and seen the tube expands first and then contracts. This is fine in two dimensions, but when I have a system of n such differential equations, 28 in the Navy intercept problem mentioned earlier, then these tubes about the true solutions are not exactly what you might think they were. The four circle figure in two dimensions, leading to the n-dimensional paradox by ten dimensions, Chapter 9, shows how tricky such imagining may become. This is simply another way of looking at what I said in earlier chapter about stable and unstable problems; but this time I am being more specific to the extent I am using differential equations to illustrate matters.
