xvi
ponents. The user may view the solution values on the interval [a, b], may
graph a ny subset of solution components on any subinterval of [a, b], and m ay
view a phase-plane graph of any so lution component versus any other so lu-
tion component on any sp ecified rectangle. Complete details for using the
six subprograms GRAPH, DIRFIELD, SOLVEIVP, POLYRTS, EIGEN, and
SOLVESYS appear in Appendix A.
The second program, PORTRAIT, so lves t he two component, auton omous
initial value problem
dy
dx =Ji (Y1, Y2); Y1 (ci) = d1i
dy
dx =f2(y1,Y2); Y2(ci) =d2ion the interval [ai , bi] where Ci E [ai, bi] for 1 :::; i :::; 10. After the solution
of an initial value problem h as been calculated, the user may elect (i) to
print the so lut ion components of any initial value problem already solved,
(ii) to graph a ny subset of the solution components previously solved in a
rectangle, (iii) to produce a phase-plane portrait of any pair of initial value
problems already solved on any rectangle, (iv) to rerun the most recent initial
value problem using a different interval of integration or init ial conditions, or
(v) to input the initial conditions for the next initial value problem to b e
solved. Complete details for using PORTRAIT appear in Appendix B.
The numerical integration procedure which is employed in the programs
SOLVEIVP, SOLVESYS, and PORTRAIT is a variable order, variable step-
size, multistep, Adams predictor-corrector method. The order is selected by
the program and varies from order one to order twelve. At each step, the step-
size is selected so that the maximum of t he relative error and the a bsolute
error remains less than 10 -^12.
[Classroom Environment[ We beli eve our "computer laboratory" is the
bes t environment in which to teach an introductory differential equations
course which emphasizes the use of computers. The center half of our la bo-
ratory h as long tables at which students sit during the lecture portion of the
class- which some days may be the entire class period. At the front of the
classroom a re two "white boards" on which lecture notes can be written and
a "smartboard." Also a t the front of the room is a microcomputer which is
linked to a projector. The instructor can demonstrate how to run programs
from t he microcomputer and discuss the output of t he programs on the smart -
board. The outside two quarters of the computer la boratory has more than
forty microcomputers with the programs CSODE and PORTRAIT on the
desktops. When the students are seated a t the microcomputers, they can run
the same program the instructor is running and compare the display on their
monitors with the display projected onto the smartboard. When the students
a re working on their own, the instructor can walk around the classroom and
assist t h e students individually.