Computer Algebra Recipes

110 CHAPTER 3. LINEAR ODE MODELS
3.1 First-Order Models
The most general nth-order linear ODE can be written in the form
a0(t) dn x(t)
dt n
+ a1(t) dn¡1 x(t)
dt n¡1
dx(t)
+ ¢¢¢+ an¡1(t)
dt + an(t) x(t) =h(t); (3.1)
the equation being labeled as linear because each term on the left-hand side is
linear, or of ¯rst order, in the dependent variable x. For the sake of de¯niteness,
the independent variable has been taken to be the time t here, but could be a
spatial variable. If h(t) = 0, the di®erential equation is said to be homogeneous,
otherwise it is inhomogeneous.
In the following two recipes, we look at two examples of inhomogeneous
¯rst-order ODEs, the ¯rst involving constant coe±cients, the second containing
variable coe±cients.
3.1.1 How's Your Blood Pressure?
Amid the pressure of great events, a general principle gives no help.
Georg Hegel, German philosopher (1770{1831)
On measuring your blood pressure, your doctor will give you two numbers.
A normal blood pressure for humans is 120/80, the ¯rst number specifying
the maximum (systolic) pressure (in units of mm of Hg) on the arterial walls,
the second number the minimum (diastolic) pressure. The blood pressure is
generated by the beating of the heart, its variation controlled by the aorta.
The aorta is the large blood vessel into which the arterial blood °ows on
leaving the heart. During the systolic phase of the heartbeat cycle, blood is
pumped under pressure from the heart into one end of the aorta, whose walls
then stretch in order to accommodate the blood. The diastolic phase then
follows during which there is no °ow of blood into the aorta, its walls elastically
contracting. The blood is then squeezed out of the aorta and around the body's
circulatory system. The following recipe presents a simple model of the variation
of blood pressure due to the beating heart and the aorta.
The entry infolevel[dsolve] will provide some information about the
dsolve command, the amount of information generally increasing as the integer
(which can vary from 1 to 5) speci¯ed on the right of the colon is increased.
> restart: infolevel[dsolve]:=2:
Let V (t) be the volume of the aorta and p(t) the pressure within it at time t.
Assuming that the aorta expands linearly with increasing p, thenV = V0 + Cp,
where V0 and C are constants. The parameter C, called the compliance, isa
measure of the stretchability of the aorta. Mentally di®erentiating this relation
produces ode1 .
> ode1:=diff(V(t),t)=C*diff(p(t),t);
ode1 := d
V (t) =C
dt
μ d
dt p(t)