The differential equation describing exponential growth is

(1) 
This can be integrated directly

(2) 

(3) 
Exponentiating,

(4) 
Defining
gives in (4), so

(5) 
The quantity in this equation is sometimes known as the Malthusian Parameter.
Consider a more complicated growth law

(6) 
where is a constant. This can also be integrated directly

(7) 

(8) 

(9) 
Note that this expression blows up at . We are given the Initial Condition
that
, so .

(10) 
The in the Denominator of (10) greatly suppresses the growth in the long run
compared to the simple growth law.
The Logistic Growth Curve, defined by

(11) 
is another growth law which frequently arises in biology. It has a rather complicated solution for .
See also Gompertz Curve, Life Expectancy, Logistic Growth Curve, LotkaVolterra Equations,
Makeham Curve, Malthusian Parameter, Survivorship Curve
© 19969 Eric W. Weisstein
19990526