1550078481-Ordinary_Differential_Equations__Roberts_

518 Ordinary Differential Equations

Clicking the OPTIONS button at the top right-hand side of the graph,
we see that the fourth option, SOLVE the SAME differential equation with
NEW conditions, will allow us to solve the same differential equation and to
change the interval of integration, the initial conditions, or both. The fifth op-
tion, SOLVE a DIFFERENT differential equation, allows to start SOLVEIVP
anew.
We did not encounter any difficulties in generating the solution to the initial
value problem y' = x - y; y(O) = 2 on the interval [-1, 4], because the
differential equation is linear-that is, y' = a(x )y + b(x )-and the functions
a(x) = -1 and b(x) = x are defined and continuous on the interval [-1, 4].

EXAMPLE Solving an Initial Value Problem

Use SOLVEIVP to calculate and graph the solution of the initial value

problem

y 1

y' = +-·

(x-l)(x+2) x'

y(-1) = 2

on the interval [-2.5, .5].

SOLUTION

To use SOLVEIVP to solve this initial value problem, we double click the

CSODE icon and when the program selection screen appears, we single click

on the SOLVEIVP button. In the box to the right of "Enter the function

f(x, y) =", we input y/((x - 1) * (x + 2)) + 1/x and press the Enter key.

Since the interval of integration is [-2.5, .5], in the box after "a= " we input

-2.5, in the box after "b = " we input .5, and then click the VERIFY a

AND b button. Since the given initial condition is y( -1) = 2, after "c = "

we input -1, after "d = " we input 2, and then click the VERIFY c AND d

AND INTEGRATE button. The Information for Graphing box appears and

we click OK. The values displayed for Xmin, Xmax, Ymin, and Ymax were

as follows: Xmin = -l.996000E + 00, Xmax = 5.000000E - 01, Ymin =

-7.538699E+OO, and Ymax = l.812798E+Ol. SOLVEIVP integrated from

-1 to the right endpoint of the interval of integration .5 and then integrated

from -1 toward the left endpoint -2.5 but stopped the integration process at

-1. 966 before reaching the left endpoint. Integration was stopped, because

one of the three events (i), (ii) or (iii) cited earlier occurred. Since we wanted

to view the graph on the interval [-2.5, .5], we changed the value of Xmin

to -2.5. Based on the computed values of Ymin and Ymax, we decided to

change the value of Ymin to -20 and the value of Ymax to 20. We clicked the

VERIFY Xmin, Xmax, Ymin, Ymax button and the options list appeared.
We wanted to see the direction field for the differential equation as well as
the graph of the numerical approximation to the solution of the initial value
problem, so we clicked the second option and then the OK button. This
caused the graph displayed in Figure A.14 to appear.