Computer Algebra Recipes

3.1. FIRST-ORDER MODELS 115
3.1.2 Greg Arious Nerd's Problem
It will be found, in fact, that the ingenious are always fanciful,
and the truly imaginative never otherwise than analytic.
Edgar Allan Poe, American writer, TheMurdersintheRueMorgue,1841
Greg Arious Nerd, an eminent mathematician at the Erehwon Institute of Technology
(EIT), has assigned the following problem for his undergraduate class to
solve using computer algebra. They are not to use dsolve to analytically solve
the relevant ODE, but instead use command structures found in the DEtools
library package that mimic the steps in a hand calculation.
Starting from rest, Evil Knievel weevil experiences a time-dependent drive
force Fdrive = t2 e ¡t andadragforceFdrag = ¡(t2 =(1 + t)) v(t), per unit mass.
Mimicking a hand calculation, determine Evil's velocity v(t) attimetand plot
it over the interval t = 0 to 10 Erehwonian time units. What distance does Evil
travel in this interval?
Here is Professor Nerd's answer key. Loading the DEtools package,
> restart: with(DEtools):
the drag and drive forces are entered.
> F[drag]:=-(t^2/(1+t))*v(t); F[drive]:=t^2*exp(-t);
Fdrag := ¡ t2 v(t)
Fdrive := t
1+t
2 e (¡t)
Applying Newton's second law, the ODE governing Evil Knievel's velocity is
> ode:=diff(v(t),t)=F[drag]+F[drive];
ode := d
dt v(t) =¡t2 v(t)
1+t + t2 e (¡t)
Although not requested, Nerd applies the odeadvisor command to ode. This
command will classify the ODE and inclusion of the help option causes a relevant
Help page with useful hyperlinks to be opened. The Help page should be
closed when one is ¯nished reading it.
> odeadvisor(ode,help); #close Help page
[ linear]
In this case, ode is classi¯ed as linear, which is obvious by inspection.
Proceeding by hand, one would look for the integrating factor2 of the ¯rstorder
inhomogeneous ODE. With Maple, the integrating factor IF for ode is
obtained by entering intfactor(ode), which is then simpli¯ed.
> IF:=intfactor(ode);
t2
IF := e
( 2 ¡ t +ln(1+t))
> IF:=simplify(IF);
R
2 f(t) dt.
For a general linear ¯rst-order ODE, _x + f(t) x = g(t), the integrating factor is e