Computer Algebra Recipes

3.1. FIRST-ORDER MODELS 111
Therateofchangeofvolume(dV=dt) of the aorta is equal to the di®erence
between the rate f(t) at which blood is pumped into the aorta by the heart and
the rate p(t)=R at which blood is pumped out of the aorta into the circulatory
system. The constant R is referred to as the systemic resistance. Thus, the
second ODE is is now given by ode2 .
> ode2:=diff(V(t),t)=f(t)-p(t)/R;
ode2 := d
p(t)
V (t) =f (t) ¡
dt R
An ODE (ode3 ) for pressure alone results on substituting ode1 into ode2 .
> ode3:=subs(ode1,ode2);
μ
d
ode3 := C
dt p(t)
= f (t) ¡ p(t)
R
As a simple model of the forcing function f(t), let's assume that during the
systolic (pumping) phase f(t) =A sin(¼t=¿), with A the amplitude and ¿ the
duration of this phase, and f(t) = 0 during the diastolic (nonpumping) phase.
So f(t) is a piecewise forcing function. If T is the time for one complete cycle
(time between heart beats), the piecewise function for two heartbeats can be
entered using the following piecewise command.
> f(t):=piecewise(t0, R>0, A>0, and P>0.
> sol:=dsolve(fode3,p(0)=Pg,p(t))
assuming (tau>0,T>tau,C>0,R>0,A>0,P>0) ;
The assuming command applies the assumption only to the command line in
which it appears. If you wish to apply assumptions to the entire work sheet,
you can use the assume command at the beginning of the work sheet. In either