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# HW8soln - lute-Hm Zeb-H-N sir-Cf 5:le this[3 Divide through...

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Unformatted text preview: lute-Hm Zeb -H-N sir-Cf, 5-:le this [3} Divide through by the coefﬁcient ofy” to obtain git—ﬁg nth—3.: y = D. The. singular points are the points where either pit} = — IT: or sit} = TEE! are discontinuous. Thus. t = i1, whm 1 _ g2 = D. m singular. {I3} Ditride through by the coefﬁoiimt of y" to obtain git-m y’+ 53:11 at = m. The singular points are the points where either pit] = _m_lm org-{t} = gig arediscnrltinuuus. 'Ihus. t = (s-+ ihr. where It is s integer. are singular points. 4;. {h} Thereis no singular points. thus the domain is [—130.00]. [:1] 'Ihesingulsr points are the points where q{t] = tanﬁtl are discontinu- nus. Thus. t = [it + ﬁlm, where h is a integer. are singular points.1:I'he domain. s mason-rm interval including 1 but not including {Jr + ‘2')": is [-itt, \$17]. 14+ If s = e'“. then s’ = —3e*3‘ and s” = 9.2-3. Thus y" + of + 3y = the-3‘ — 12e'3‘+ Ee'a' = 0. 1ft! = B'hﬂmy’ = —e" and y" = rt'thus yt+w+e=e~=—e'=+ae-‘=U- —at —t ' 0D the Ether hand. W[e_3',e."] = deg [ "33-31 :24 1 = 25—“ 5g '1 Therefore 3 is a tirndamental set of solutions. The general solution is y = cue—3" + me" 15. Ify=t+1.theny"=1andy"=ﬂ.ffy=t-1.theng’=1arld y”=ﬂ.0ntl:ieotherhaltd.Wit+1,t—1]=det [ “1'1 t111=2¢n ThereforeiS' is a ﬁmdamental set of solutions. The general solution is s=ciit+ll+ezit-1} 1t. Ity=1.iheny'=oahu(ty’3’=o. Ify=ln{t). rhehy’=1,iiahu [iy'r = {1)' = o. ohmehuerhand. _ 1 left} _ W[1,1n{t}] _det [ [a 1ft ]—- 1ft a-ED Therefore .5 is a fundamental set of solutions. The general solution is y=e1+eslnitl Meet—k 29D Hi0¢\$ Huﬁma r‘é‘: L l. The characteristic equation is s1 — 4 = D. The characteristic roots are :2. The general solution is y = Clem + (Tie-2'. T. The characteristic equation is s2 + 43 + 4 = i]. There is a double characterisu'c root. —2. The general solution is y = e'WC-‘lr + 02}. 16. The characteristic equation is s2 + 3+ 1 = (i. The characteristic roots are —§ i 131'. The general solution is y = e"‘4" 2(«117:1 cospéir} + 02 sin{i?t}}. ...
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