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Jim Lambers
MAT 285
Spring Semester 2012-13
Lecture 1 Notes
These notes correspond to Section 1.1 in the text.
Some Basic Mathematical Models
This course is an introduction to differential equations, which are equations that depend on derivatives of unknown quantities. Differential equations arise in mathematical models of a wide variety
of phenomena, such as propagation of waves, dissipation of heat energy, population growth, or
motion of fluids. Solutions of differential equations yield valuable insight about such phenomena,
and therefore techniques for solving differential equations are among the most essential methods of
applied mathematics.
We now illustrate mathematical models based on differential equations. Newton’s Second Law
F = ma = m ,
where F , m, a and v represent force, mass, acceleration and velocity, respectively. We use this
law to develop a mathematical model for the velocity of a falling object that includes a differential
equation. The forces on the falling object include gravity and air resistance, or drag; to simplify
the discussion, we neglect any other forces.
The force due to gravity is equal to mg, where g is the acceleration due to gravity, and the
drag force is equal to −γv, where γ is the drag coefficient. We use downward orientation, so that
gravity is acting in the positive (downward) direction and drag is acting in the negative (upward)
direction. In summary, we have
F = mg − γv.
Combining with Newton’s Second Law yields the differential equation
= mg − γv
for the velocity v of the falling object.
Another example of a mathematical model is a differential equation for the population p of a
species, which can have the form
= rp − d,
where the constant r is the rate of reproduction of the species. In general, r is called a rate constant
or growth rate. The constant d indicates the number of specimens that die per unit of time, perhaps
due to predation or other causes.
In general, a mathematical model is constructed by identifying all of the relevant quantities
and how they relate to one another. Associating variables with these quantities facilities the
development of an equation that relates them. Such an equation can be obtained from, for example,
physical laws that govern the behavior of these quantities.