On Tue, Dec 06, 2005 at 07:32:21PM +1100, Alasdair McAndrew wrote:
> 5) ODE's with an extra parameter. My first year calculus students cut their
> teeth on the sort of simple DE's which contain a proportionality constant
> and two initial conditions:
>
> dy/dx = ky(1-y), y(0)=1, y(10)=2
>
> for example. So they need to determine k as well as the constant of
> integration. I've been experimenting, but have yet to find an easy natural
> way of solving such equations.
I've found that students learn at lot by looking at the geometry of
the problem, even before they learn analitical techniques.
I think your example has no answer, but in any case, what I propose is
something like this (maxima 5.9.2):
load("plotdf");
plotdf(k*y*(1-y),[trajectory_at,0,1],[parameters,"k=1"],[sliders,"k=0.1:5"],[direction,forward]);
the student can then move the slider and conclude that, independently of the
value of k (at least in the interval (0.1,5)), the solution will always
end up in the stable equilibrium point y(10)=1.
Regards,
Jaime