xvIn Chapter 10 we present techniques for determining the b ehavior of so-
lutions to systems of first-order differential equations without first finding
the solutions. To this end, we define and discuss equilibrium points (criti-
cal points), various types of stability and instability, and phase-plane graphs.
Next, we show how to use computer software to solve systems of first-order
differential equations numerically, how to graph the solution components and
how to produce phase-plane graphs. We also state stability theorems for sys-
tems of first-order differential equations. Throughout this chapter we develop
and discuss a wide variety of models and applications which can be written
as vector init ial value problems and then so lved numerically.
Although not the primary purpose for which this book was intended, it
may also be used as (1) a text for a second course in ordinary differential
equations for students who have taken a traditional course previously, (2) a
text for a laboratory course , and (3) a supplement to the differential equations
component of a numerical analysis course.jcomments on Our Computer Software I No prior knowledge of computers
or of a ny particular programming language is required to use our computer
software. Furthermore, no programming can be done. The user simply selects
a program to perform a particular task and enters the appropriate data. Then
the user interacts with the program by selecting options to be executed. The
user only needs to know the acceptable formats for entering numerical data
a nd the appropriate syntax for entering functions. The computer disc included
with this text contains two main programs.
The first program, CSODE, includes the six subprograms: GRAPH, DIR-
FIELD, SOLVEIVP, POLYRTS, EIGEN, and SOLVESYS. The subprogram
GRAPH graphs a function y = f ( x) on a specified rectangle in the xy-plane.
The instructional purposes of this program are to teach the user how to en-
ter functions into programs properly and how to interact with programs.
Of course, GRAPH may be used to graph expli cit solut ions of differential
equations and view their behavior. The subprogram DIRFIELD graphs the
direction field of the first-order differential equation y' = f ( x, y) on a sp ec-
ified rectangle. The output of DIRFIELD permits the user to "see" where
the differential equation is and is not defined, where solutions increase and
decrease, and where extreme values occur. Sometimes asymptotic behavior
of the solutions can be determined also. SOLVEIVP solves the scalar first-
order initial value problem y' = f(x, y); y(c) =don an interval [a , b] where
c E [a , b]. The solution values Yi at 1001 equ ally spaced points Xi E [a, b]
may be viewed or plotted, with or without the associated direction field, on
a rectangle specified by t he user. The subprogram POLYRTS calculates t he
roots of a polynomial with complex coefficients of degree less than or equ al
to ten. EIGEN calculates the eigenvalues and associated eigenvectors of a n
n x n matrix with real coefficients where 2 :::; n :::; 6. The sixth subprogram
SOLVESYS solves the vector initial value problem y' = f(x, y); y(c) =don
t he interval [a , b] where c E [a , b] and the vector has from two to six com-