34
Economic Dynamics
present at time
t
. If
dn
/
dt
denotes the number of atoms that degenerate per unit of
time, then according to Rutherford
dn
dt
= −
λ
n
λ >
0
(2.9)
where
λ
is the decay constant of the substance concerned and
n
is a continuous
function of time. This is a firstorder linear homogeneous differential equation and
is identical in form to the exponential population growth specified in example 2.6
above. We shall return to this example later when we consider its solution and how
the solution is used for calculating the
halflife
of a radioactive substance and how
this is used to authenticate paintings and such items as the Turin shroud.
Example 2.8
In this example we consider a continuous form of the Harrod–Domar growth
model. In this model savings,
S
, is assumed to be proportional to income,
Y
;
investment,
I
, i.e., the change in the capital stock, is proportional to the change in
income over time; and in equilibrium investment is equal to savings. If
s
denotes
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 Fall '11
 Dr.Gwartney
 Economics, Derivative, vy, homogeneous differential equation

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