Explain the determine red 6.3.3 Ехample CThe equationYk+1Yk – 2yk = -2(6.38)can be transformed into the linear equationXk+2 – 2xk+1+2xk = 0(6.39)by means of the substitutionYk =Xk+1/xk.(6.40)The characteristic equation for equation (6.39) isp2 – 2r + 2 = 0,(6.41)and its two complex conjugate roots arer1,2 = 1+i = v2e±i™/4_(6.42)Therefore, the general solution of equation (6.39) isX} = 2k/2[D1 cos(tk/4)+ D2 sin(Tk/4)],(6.43)andD1 cos[T(k+ 1)/4] + D2 sin[r(k +1)/4]Dị cos(Tk/4) + D2 sin(tk/4)Yk =V2-(6.44)200Difference EquationsNow, define the constant a such that in the interval -T/2 < a < T/2,tan a =D2/D1.(6.45)With this result, equation (6.44) becomesV2 cos[(T/4)(k + 1) – a]Yk =cos(Tk/4 – a)(6.46)orYk1- tan(ak/4 – a).(6.47)This is the general solution to equation (6.38).Note that since tan(0+ 7) = tan 0, the solution has period 4; i.e., for givenconstant a, equation (6.47) takes on only four values; they areYo = ]tan(-a),- tan(7/4 – a),1- tan(7/2 – a),1– tan(37/4 – a).Y1 =1-(6.48)Y2 =Y3An easy calculation shows that yo = Y4.

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6.3.3 Example C The equation Yk+1Yk – 2yk = -2 (6.38) can be transformed into the linear equation Xk+2 2xk+1 + 2xk = 0 (6.39) by means of the substitution Yk = *k+1/xk. (6.40) The characteristic equation for equation (6.39) is p2 – 2r + 2 = 0, (6.41) and its two complex conjugate roots are r1,2 = 1+i = V2etin/4 (6.42) Therefore, the general solution of equation (6.39) is æk = 26/2[D1 cos(Tk/4)+ D2 sin(tk/4)], (6.43) and - 15D1 cos[T(k+1)/4] + D2 sin[T(k+1)/4] Dị cos(Tk/4) + D2 sin(rk/4) (6.44)

6.3.3 Example C The equation Yk+1Yk – 2yk = -2 (6.38) can be transformed into the linear equation Xk+2 2xk+1 + 2xk = 0 (6.39) by means of the substitution Yk = *k+1/xk. (6.40) The characteristic equation for equation (6.39) is p2 – 2r + 2 = 0, (6.41) and its two complex conjugate roots are r1,2 = 1+i = V2etin/4 (6.42) Therefore, the general solution of equation (6.39) is æk = 26/2[D1 cos(Tk/4)+ D2 sin(tk/4)], (6.43) and - 15D1 cos[T(k+1)/4] + D2 sin[T(k+1)/4] Dị cos(Tk/4) + D2 sin(rk/4) (6.44)

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section11.4: Plane Curves And Parametric Equations

Problem 44E

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