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gait gt | fanct: Qs ~ stating the wronskien of 4,/6), 42 CE Jed f = 9,(e) gol) = J ] 4. Ge) D2 oO ao Ss >, yh = 2e2* sin) + C2 cos €) Dt fT) ly’ [22% fon de) + 2 Isinf Tob th) — eft of-)_ tgnlelos(t) — (i swid)+ ave) )= |-ebt 3) Determine | the | LMntervel ta which anes TNB ts Certain tolhave la! unique- twice! oi Plaeatiable golds b NOT | a Bind! the Solution: (x-2) Phe yl ple) 1S Conkinuevsy every where , excep + xoo not defmeol in this indervel ; Lareesh interval is | xe C ad Ww Yla a a que. solubian By the existence amd uniqueness, theorem, there isa un foc the! LYP) tn this intervel- lugging in y, CeJand Ye Le) tnto the DE_, we cert ensily. see they are tndeed solutions. y,le)oe*, jy, a 2er* y,"2 yer* yaléda tem, ya's Dtelts PF) a Me be etts ber Tc e3fthe fev o Liadake ef coluttan we Ste theit vu eat fl, y at ttt wy. 4, ] led = = et* (9 ett ett) — L.e2# 9. e2t HPS ea Let Ft, rt = eft Smee et is non-tero 2 w avicled so are ) ao undamertel set of solitons andthe whele antera- 0 § hi + 3 t so] 4 ie ak: racers t3 > d rr. ; ot gal 3 ca se Ces ce ¥ ; Ea § uw 8 ie ul Pa) + += > mle 3 < te jo f ~ 5 \ Ar mr OD P {@) w 1) f aR R 3 eo ei QO Als J a a J] + a d ?rs y =~ a + atte | je fle ecu Or SO Se TS a if oJ a! | Sy Cal ES Ey st D2 Pay LLU lsh lg | y i my ™~ b Fal Sq 6 isp! IA cy 23 : a ah Agtits 3 g OM & Like [| te [NY othe given de; SO — ye -lFi Thus (eda elt. ot cette e+ Cosle)sisme)) Ano fen L solution: yea e tC, cosle) + Cp sinle)) Now using the given y On , 4 (Aly a Jie a (CO CoslTh) + oom (Xlu)) = a, & lees) = 2. ae Vn a Es ip a ats Hencel:| Cyeqarhe™4 | Ales: y (Rly) =e" Ceycos (Ay) 4ersm Xl) ) + a4 (ey ean" ly) + Co CogCKh)) = ek : (-G-c,) =O ==> Ciao [ea nate Aly t —»| 4(O)= 2Ge sine) = a= 2. Pe | a 4b)! Gasider Shel T.VP: Sut-ei2uw4+2uso | vle)tal u'ea2 ule) of this probes. | st, lsucln +h a) The characterisHe eq. —> Berta t=0 , 2s eF thas, +he generol solution _u (#)= at. Ca Ons Bt ) + so lSE4)) Solving foc tartial| conditions: ulo)=2=\e°. Ce, Cosco) + cp -sinlo) mic, Ls [ea 2 u'lo) = Zot. Cos Coslo)+ peceny ar iS CBiscrsinb) « fai C Coste) ) aH a] | @ 2 Pos C (BL Co. we) => ¢cCg> CR “Here the solation :| uled= eM C2cosl Be) + ein CE 4) By b) Useag a graph we.con see that after Ta d1b-75, we have |ule)lco.d. Ll Froblenl aie" Fy st k el-| val bn le) bocsame x x ee (a dt (4 A? 7] Re Now for _y (+); by cham cule dy dg (Ae || 4 [ds at ax Lae tds! ay A Nita we oa a 2 = : + eal = a _ dj)! att ) Ae” dx ee abet 2) a iz Now inthe eg, grven, Paktmng ux) instead of 4 le) : 2 WG)4 seve): Bye) ee 4 (45 a9!) oe ida List So oOo (6) gd t Vode ada? ae = ye) tq x) + dyl Jeo, Now seve very charaatesishe 2, Fel LP le Bl lato [iC ddd | ek | coe 8 And the isolation: y(x) = Qe i+ Cp ee, Bat smee xebn (4) ; y le = Ca¥ + Co. 473 
