Week 7 Lecture Notes, Lec01
9.1. Differential Equations
In general, a differential equation is an equation that contains an unknown function
and one or more of its derivatives. The order of a differential equation is the order
of the highest derivative that occurs in the equation.
Example.
? + ?
′
+ ?
′′
= ?
Differential equations can be used to create mathematical models for real-life
phenomena.
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Population Growth
It is known that the population grows at a rate proportional to the size of the
population. If we define population, P, as a function of time, then we would have:
?𝑃
?𝑡
= 𝑘𝑃(𝑡)
Where k is a constant, and
?𝑃
?𝑡
is the rate of growth of the population.
Assuming there is enough nutrition, there
’
s no epidemic diseases and no predators,
the population will be always increasing;
?𝑃
?𝑡
> 0
. In addition, the population itself is
a positive number; so
𝑘 > 0
.
This equation,
?𝑃
?𝑡
= 𝑘𝑃(𝑡)
, asks us to find a function whose derivative is a constant
multiple of itself. A possible form of such function would be
𝑃(𝑡) = 𝐶?
𝑘𝑡
.
𝑃′(𝑡) = 𝑘(𝐶?
𝑘𝑡
) = 𝑘𝑃(𝑡)