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A wide variety of natural phenomena such as projectile motion, the flow of electric current, and the progression of chemical reactions are well described by equations that relate changing quantities. As the derivative of a function provides the rate at wh€7,57
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A wide variety of natural phenomena such as projectile motion, the flow of electric current, and the progression of chemical reactions are well described by equations that relate changing quantities. As the derivative of a function provides the rate at wh
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HUT301
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Kerala Technical University
A wide variety of natural phenomena such as projectile motion, the flow of electric current,
and the progression of chemical reactions are well described by equations that relate changing
quantities. As the derivative of a function provides the rate at which that function is changing
with respec...
1 Overview
A wide variety of natural phenomena such as projectile motion, the flow of electric current,
and the progression of chemical reactions are well described by equations that relate changing
quantities. As the derivative of a function provides the rate at which that function is changing
with respect to its independent variable, the equations describing these phenomena often
involve one or more derivatives, and we refer to them as differential equations. In these
notes we consider two important aspects in the theory of ordinary differential equations:
1. Developing models of physical phenomena; and 2. Determining whether our models are
mathematically “well-posed” (do solutions exist? are these solutions unique? do the solutions
we find for our equation genuinely correspond with the phenomenon we are modeling).
1
, Solutions to ordinary differential equations cannot be determined uniquely without some
outside condition, typically an initial value or a boundary value. In order to understand the
nature of this information, consider the general first order equation
y ′ = f (t, y), (1.1)
for which ′ denotes differentiation with respect to t. Assuming f (t, y) is sufficiently dif-
ferentiable, we can develop a solution to (1.1) for t sufficiently small through the Taylor
approximation,
1
y(t) = y(0) + y ′ (0)t + y ′′(0)t2 + .... (1.2)
2
Suppose we know the initial value for y(t), y(0). Observe, then, that we can compute y ′(0)
directly from (1.1):
y ′(0) = f (0, y(0)).
Similarly, by differentiating (1.1) with respect to t, we have
∂ ∂
y ′′ = f (t, y) + f (t, y)y ′,
∂t ∂y
and we can compute y ′′(0) as
∂ ∂
y ′′ (0) = f (0, y(0)) + f (0, y(0))y ′(0).
∂t ∂y
Proceeding similarly, we can develop the entirety of expansion (1.2).
2 Compartment Analysis
Suppose y(t) denotes the amount of substance in some compartment at time t. For example,
y(t) might denote the liters of gasoline in a particular tank or the grams of medicine in a
particular organ. We can compute the change in quantity y(t) in terms of the amount of
this quantity flowing into the compartment and the amount flowing out, as
dy
= input rate − output rate.
dt
Example 2.1. Suppose saltwater is pumped into a tank with constant rate r cm3 /s, and is
pumpted out at the same rate, and that the concentration of salt in the incoming water is
c grams/cm3 . If the volume of water in the tank is V cm3 , find an equation for the amount
of salt in the tank at time t.
Let y(t) denote the grams of salt at time t, and notice that
dy y(t) y(t)
= input rate − output rate = cr − r = r(c − ).
dt V V
2
, Example 2.2. (Drug concentration in an organ.) Suppose blood carries a certain drug into
an organ at variable rate rI (t) cm3 /s and out of the organ with variable rate rO (t) cm3 /s,
and that the organ has an initial blood volume V cm3 . If the concentration of drug in the
body is c(t) g/cm3 , determine an ODE for the amount of drug in the organ at time t.
Let y(t) denote the amount of drug in the organ at time t, measured in grams. The input
rate is then rI (t)c(t), while the output rate, assuming instantaneous mixing, is Vy(t) r (t),
(t) O
where the volume of blood in the organ V (t) can be computed as the initial volume V plus
the difference between the blood that flows into the organ over time t and the blood that
flows out during the same time:
Z t
V (t) = V + rI (s) − rO (s)ds.
0
We have, then, the ODE
dx y(t)
= c(t)rI (t) − Rt rO (t).
dt V + 0 rI (s) − rO (s)ds
△
Example 2.3. Suppose M grams of a certain heart medication are injected into a patient
at time 0, and that whenever the drug is present in the heart its absorption rate out of
the bloodstream (and into the heart tissue) is proportional to the amount in the heart with
proportionality constant rA s−1 . If blood flows into the patient’s heart with rate rI cm3 /s
and out with rate rO cm3 /s, and if the volume of blood in the heart at time 0 is VH and the
volume of blood in the patient’s body (minus the heart) at time 0 is VB , develop a model
for the amount of drug absorbed into the heart tissue by time t.
Let y(t) denote the amount of drug in the heart at time t, and let A(t) denote the total
amount absorbed into the heart tissue by time t. Notice that dA dt
= rA y(t) (i.e., the rate of
dA
absorption dt is proportional to the amount in the heart), and so
Z t
A(t) = rA y(s)ds.
0
Now,
dy M − y(t) − A(t) y(t)
= Rt rI (t) − Rt rO (t) − rA y.
dt VB − 0 rI (s) − rO (s)ds VH + 0 rI (s) − rO (s)ds
Coupling this with
dA
= rA y,
dt
we have a system of two equations for the quantities y(t) and A(t).
Example 2.4. (Cleaning the Great Lakes.) The Great Lakes are connected by a network
of waterways, as roughly depicted in Figure 2.1. Assume the volume of each lake remains
constant, and that water flows into the lake with volume Vk at rate rk . Suppose pollution
3
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