HW__6_Solution

# HW__6_Solution - 55 3/3 HW é So/u‘i't‘cms‘ ¢...

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Unformatted text preview: 55 3/3 HW# é So/u‘i't‘cms‘ ¢ 3.9—3 (1:2 — 215 + 2),,[k] = o The characteristic equation is 12 - 21 + 2 = (1 - 1 -j1)(1 — 1 +j1) = 0. The roots are 1:l:jl = ﬁe*”"‘. yUc] = C(ﬁ)* cos( fl: + 9) Setting I: = —1 and —2 and substituting initial conditions yields 1 = ﬁcos(-§ +0) = :52.(7‘2-cos9 + i—sina) = §cos(—§.+ 0) = §sin 9 Solution of these two simultaneous equations yields ccos9=2 c=2 =9 csin9=0 0:0 ‘ lytgk1= 2(ﬁ)*cos(%k) (Prob . 2. 9 Characteristic equation is 72 +21 + 2 = (7 +1 —j1)(‘Y +1+j1)= 0. The roots are -1:l:j1= ﬂei’s'“. y[k] = c(x/§)’f cos(37"k + 9) Setting 1: = 0, 1, and substituting initial conditions yields ' 0=ccos€ c=2 => 2=\/§ccos(JT”+9) 9=—% We] = 20/5)" c0507” ‘ g) = 205),: ““07”“ gm Erplw gg 3/3 Hw #4 5°!“ Pmb.3 & _ I W: +2] +4y[lc] = 0 0! (E’ + 4)y[k1 = 0 The characteristic equation is 72 + 4 = (7 + j2)(7 — 12) = 0. The roots axe :tj2 = 231*”. yUc] = ca)“ cos(§k + 9) Setting 1: = 0, 1, and substituting initial conditions yields 1=ccoso I } cc050=1} c=ﬁ => => 2 = 2ccos I +0) = —2csin0 csino = -1 2 ytk1= ﬁi2>*c_os(%h - a 9-1“ ...
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