ODE Lecture Notes
Section 1.1
Page 1 of 6
Section 1.1: Some Basic Mathematical Models; Direction Fields
Big Idea: A direction field is a useful tool for visualizing the behavior of a first-order differential
equation.
Big Skill: You should be able to develop some mathematical models in terms of differential
equations, and generate a direction field for a given differential equation.
A differential equation is an equation containing derivatives.
A few applications of differential equations:
 Fluid flow
 Current flow in an electric circuit
 Heat flow
 Seismic wave propagation
 Electromagnetic wave propagation
 Population dynamics
A mathematical model is an equation that attempts to describe some physical process.
Practice:
1. Formulate a mathematical model in the form of a differential equation to describe an
object falling vertically through the atmosphere near sea level. Approximate the air drag
as being directly proportional to the object’s velocity. Take the positive y direction as
pointing down.
ODE Lecture Notes
Section 1.1
Page 2 of 6
A direction field (or a slope field) is a graph consisting of short line segments that indicate the
slopes of the tangent lines to the family of curves of a differential equation at a discrete number
of points. The angle of the segments are computed by the value of the derivative at the discrete
points. Direction fields are useful because you can “trace” the solution of the equation by
following the arrows.
Example:

Create a direction field in Winplot for the differential equation
dy
 2 y  y  2 .
dx
ODE Lecture Notes
Section 1.1
Page 3 of 6
Example:

dy
 2 y  y  2 .
dx
Type the differential equation in the ODE Editor window, then click Enter:

Modify the top right graph by right-clicking on it and then choosing Edit…:

Modify the scales of the graph (and the size of the window) to get desired viewing area:

Create a direction field in ODE Architect for the differential equation
ODE Lecture Notes
Section 1.1
Page 4 of 6
Practice:
2. Using a mass of 10 kg and an air drag proportionality constant of 2 kg/s, draw a few
segments of the slope field for the previous free-fall problem by hand, and then generate
a slope field using Winplot and ODE Architect. Discuss the behavior of the object for
different initial values of v.
t
v
dv
dt
ODE Lecture Notes
Section 1.1
Page 5 of 6
An equilibrium solution is a solution to a differential equation that does not change (i.e., is a
constant). Equilibrium solutions are found by setting y = 0 (because the derivative of a constant
is zero) and solving the resulting equation.
A stable equilibrium solution is a solution that other solutions tend to converge toward.
Direction field lines point toward stable equilibriums.
An unstable equilibrium solution is a solution that other solutions tend to diverge away from.
Direction field lines point away from unstable equilibriums.
Practice:
3. Find the equilibrium solution for the free-fall problem, and comment on how it appears
on the direction field of the differential equation.
dy
 2 y  y  2  , and based on the direction field,
dx
determine the behavior of y as x   , including any dependency on the value of y(0).
4. Find the equilibrium solutions of
ODE Lecture Notes
Section 1.1
Page 6 of 6
5. Determine the behavior of the solutions of y  2  t  y as t   , including any
dependency on the value of y(0).
More comments on direction fields:




dy
 f t, y  .
dt
f is sometimes called the rate function (think of the exponential growth differential
dy
 ky , where k is the growth rate (or rate costant)).
equation
dt
Direction fields are nice because they do not require solving the differential equation.
Use a computer to do the work!
Direction fields are useful in studying equations of the form
Constructing Mathematical Models
 Identify the independent and dependent variables; often time is the independent variable.
 Choose units of measure for the variables.
 Articulate the basic principle underlying the problem.
 Express that principle in terms of the variables you chose.
 Make sure that every term has the same physical units (i.e., your equation is
dimensionally consistent).
 For this and the next chapter, you should only need one equation to describe the
principle.
One last practice:
6. Create a model for a mouse population that grows at a rate of 50% per month, and loses
15 mice per day due to predation.