(i) On rectangular axes, plot yk+1 as a function of yk using the functional relation given by equation (2.118). Also plot on this same graph the straight line yk+1 = Yk· (ii) For any initial value yo, draw a vertical line from the horizontal (yk) axis, with this value, to the curve yk+1= f(yk); this gives y1.
(i) On rectangular axes, plot yk+1 as a function of yk using the functional relation given by equation (2.118). Also plot on this same graph the straight line yk+1 = Yk· (ii) For any initial value yo, draw a vertical line from the horizontal (yk) axis, with this value, to the curve yk+1= f(yk); this gives y1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
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Explain the determain and illustrate using the drawing and this complete
Transcribed Image Text:2.7
A GENERAL FIRST-ORDER EQUATION:
84
GEOMETRICAL METHODS
In this section, we will examine, by means of geometrical techniques, the
possible solution behaviors of the differece equation
Yk+1 = f(yk),
(2.118)
where, in general, f(yk) is a nonlinear function of yk. Our goal is to study the
behavior of solutions of this equation for arbitrary initial values as k → .
Note that given any initial value yo, equation (2.118) can be used to obtain
Y1, which can then be used to obtain y2, etc. Thus, to each initial value yo,
there corresponds a unique sequence (yo, Y1, Y2, Y3, ...). This procedure can be
described graphically in the following manner:
(i) On rectangular axes, plot Yk+1 as a function of yk using the functional
relation given by equation (2.118). Also plot on this same graph the
straight line Yk+1 = Yk•
(ii) For any initial value yo, draw a vertical line from the horizontal (yk)
axis, with this value, to the curve yk+1
f(yk); this gives y1.
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