Explain the determain and illustrate using the drawing and this complete 2.7A GENERAL FIRST-ORDER EQUATION:84GEOMETRICAL METHODSIn this section, we will examine, by means of geometrical techniques, thepossible solution behaviors of the differece equationYk+1 = f(yk),(2.118)where, in general, f(yk) is a nonlinear function of yk. Our goal is to study thebehavior 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 obtainY1, 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 bedescribed graphically in the following manner:(i) On rectangular axes, plot Yk+1 as a function of yk using the functionalrelation given by equation (2.118). Also plot on this same graph thestraight 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+1f(yk); this gives y1.

Plot yk+1 as a function of yk using the given function.

(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.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.6: The Inverse Trigonometric Functions

Problem 92E

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