Elmn Diffnil Equions nd Bound Vlu Poblms Bo. & DiPim, 9 h Ediion Chp : Sond Od Diffnil Equions 6 คณ ตศาสตร ว ศวกรรม สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา /555 ผศ.ดร.อร ญญา ผศ.ดร.สมศ กด วล ยร ชต
Topis Homognous Equions-Consn Coffiins i Fundmnl Soluions of Lin Homognous Equions Complx Roos of Chisi Equion Rpd Roos; Rduion of Od Nonhomognous Equions; Mhod of Undmind Coffiins Viion of Pms
.: nd Od Lin Homognous Equions- Consn Coffiins Elmn Diffnil Equions nd Bound Vlu Poblms, 9 h diion, b Willim E. Bo nd Rihd C. DiPim, 9 b John Wil & Sons, In. A sond od odin diffnil quion hs h gnl fom f,, wh f is som gin funion. This quion is sid o b lin if f is lin in nd ': g p q Ohwis h quion is sid o b nonlin. A sond od lin quion ofn pps s P Q R G If G= = fo ll, hn h quion is lld homognous. Ohwis h quion is nonhomognous.
Homognous Equions, Iniil Vlus In Sions.5 nd.6, w will s h on soluion o homognous quion is found, hn i is possibl o sol h osponding nonhomognous quion, o ls xpss h soluion li in ms of n ingl. i Th fous of his hp is hus on homognous quions; nd in piul, hos wih onsn offiins: b W will xmin h ibl offiin s in Chp 5. Iniil ondiions pill k h fom, Thus soluion psss hough,, nd slop of soluion, is qul o '.
Exmpl : Infinil Mn Soluions of Consid h sond od lin diffnil quion Two soluions of his quion, Oh soluions inlud, 5, 5 5 Bsd on hs obsions, w s h h infinil mn soluions of h fom I will b shown in Sion. h ll soluions of h diffnil quion bo n b xpssd in his fom. 5
Exmpl : Iniil Condiions of Now onsid h following iniil lu poblm fo ou quion:,, W h found gnl soluion of h fom Using h iniil quions, Thus / / /, / 6
Exmpl : Soluion Gphs of Ou iniil lu poblm nd soluion,, / / Gphs of boh nd gin blow. Obs h boh iniil ondiions sisfid. ' / / ' / /.5..5..5..5. 7
Chisi Equion To sol h nd od quion wih onsn offiins, b w bgin b ssuming soluion of h fom =. Subsiuing his ino h diffnil quion, w obin b Simplifing, i nd hn b b This ls quion is lld h hisi quion of h diffnil quion. W hn sol fo b foing o using qudi fomul. 8
Gnl Soluion Using h qudi fomul on h hisi quion b, w obin wo soluions, nd. Th h possibl suls: Th oos, l nd. Th oos, l nd =. Th oos, omplx. b b In his sion, w will ssum, l nd. In his s, h gnl soluion hs h fom 9
Iniil Condiions,, l, Fo h iniil lu poblm Fo h iniil lu poblm w us h gnl soluion,,, b w us h gnl soluion ogh wih h iniil ondiions o find nd. Th is,, Sin w ssuming, i follows h soluion of h fom = o h bo iniil lu poblm will lws if f i ii l dii xis, fo n s of iniil ondiions.
Exmpl Gnl Soluion Consid h lin diffnil i quion 5 6 Assuming n xponnil soluion lds o h hisi quion: 5 6 Foing h hisi quion ilds wo soluions: = - nd = - Thfo, h gnl soluion o his diffnil i quion hs h fom
Exmpl Piul Soluion Consid h iniil lu poblm, 5 6, Fom h pding xmpl, w know h gnl soluion hs h fom: Wih dii: ' Using h iniil ondiions: Thus 7 9, 7. 5.5 9 7 5.5.5. 9 7..5..5..5
Exmpl : Iniil Vlu Poblm Consid h iniil lu poblm 8,, / Thn 8 Foing ilds wo soluions, = / nd = / Th gnl soluion hs h fom / / Ui Using iiil iniil ondiions: dii / / Thus / / / / 5/, / 5/ / / / 5/.5..5..5
Exmpl 5: Find Mximum Vlu Fo h iniil i i lu poblm in Exmpl, o find h mximum lu ind b h soluion, w s = nd sol fo : 6 9 7 8 7 / 6 ln 7 /.5. 6 7 s 5.5..5..5 9 7..5..5..5
.: Fundmnl Soluions of Lin Homognous Equions Elmn Diffnil Equions nd Bound Vlu Poblms, 9 h diion, b Willim E. Bo nd Rihd C. DiPim, 9 b John Wil & Sons, In. L p, q b oninuous funions on n inl I =,, whih ould b infini. Fo n funion h is wi diffnibl on I, dfin h diffnil opo L b L p q No h L[] is funion on I, wih oupu lu L Fo xmpl, p L p q, q, sin os sin sin, I, 5
Diffnil Opo Noion In his sion w will disuss h sond od lin homognous quion L[] =, long wih iniil ondiions s indid blow: L p, q - - - - - - - W would lik o know if h soluions o his iniil lu poblm, nd if so, h uniqu. Also, w would lik o know wh n b sid bou h fom nd suu of soluions h migh b hlpful in finding soluions o piul poblms. Ths qusions ddssd in h homs of his sion. 6
Thom.. Exisn nd Uniqunss Consid h iniil lu poblm p q g, wh p, q, nd g oninuous on n opn inl I h onins. Thn h xiss uniqu soluion = on I. 7
Exmpl p q g, Consid h sond od lin iniil lu poblm ',, Wing h diffnil quion in h fom : ' ' p ' q g p /, q / nd g Th onl poins of disoninui fo hs offiins = nd =. So h longs opn inl onining h iniil poin = in whih ll h offiins oninuous is < < Thfo, h longs inl in whih Thom.. guns h xisn of h soluion is < < 8
Thom.. Pinipl of Supposiion If nd soluions o h quion L[ ] p q hn h lin ombinion + is lso soluion, fo ll onsns nd. To po his hom, subsiu + in fo in h quion bo, nd us h f h nd soluions. Thus fo n wo soluions nd, w n onsu n infini fmil of soluions, h of h fom = +. Cn ll soluions n b win his w, o do som soluions h diffn fom logh? To nsw his qusion, w us h Wonskin dminn. 9
Th Wonskin Dminn of Suppos nd soluions o h quion L[ ] p q ------- Fom Thom.., w know h = + is soluion o his quion. Nx, find offiins suh h = + sisfis h iniil ondiions -------, To do so, w nd o sol h following quions: ------- -------
Th Wonskin Dminn of Soling h quions w obin Soling h quions, w obin In ms of dminns:,
Th Wonskin Dminn of In od fo hs fomuls o b lid h dminn W in In od fo hs fomuls o b lid, h dminn W in h dnomino nno b zo: W W, W W is lld h Wonskin dminn, o mo simpl, h Wonskin of h soluions nd. W will somims us h noion W somims us h noion, W
Thom.. Suppos nd soluions o h quion L [ ] p q wih h iniil ondiions, Thn i is lws possibl o hoos onsns, so h sisfis h diffnil quion nd iniil ondiions if nd on if h Wonskin W is no zo h poin
Exmpl In Exmpl of Sion., w found h nd w soluions o h diffnil quion 5 6 Th Wonskin of hs wo funions is W 5 Sin W is nonzo fo ll lus of, h funions nd n b usd o onsu soluions of h diffnil quion wih hiiil iniil ondiions dii n lu of
Thom.. Fundmnl Soluions Suppos nd soluions o h quion L[ ] p q. Thn h fmil of soluions = + wih bi offiins, inluds soluion o h diffnil quion if n onl if h is poin suh h W,,. Th xpssion = + is lld h gnl soluion of h diffnil quion bo, nd in his s nd sid o fom fundmnl s of soluions o h diffnil quion. 5
Exmpl : fundmnl s of soluions? Consid h gnl sond od lin quion blow, wih h wo soluions indid: p q Suppos h funions blow soluions o his quion:,, Th Wonskin of nd is W fo ll. Thus nd fom fundmnl s of soluions o h quion, nd n b usd o onsu ll of is soluions. Th gnl soluion is 6
Exmpl 5: Soluions of Consid h following diffnil quion: Consid h following diffnil quion: Show h h funions blow fundmnl soluions:, To show his, fis subsiu ino h quion: /, / / / / Thus is indd soluion of h diffnil quion. Simill, is lso soluion: 7
Exmpl 5: Fundmnl Soluions of Rll h /, To show h nd fom fundmnl s of soluions, w lu h Wonskin of nd : W / / / / / Sin W fo >,, fom fundmnl s of soluions fo h diffnil quion, 8
Thom..5: Exisn of Fundmnl S of Soluions Consid h diffnil quion blow, whos offiins p nd q oninuous on som opn inl I: L [ ] p q L b poin in I, nd nd soluions of h quion wih sisfing iniil ondiions, nd sisfing iniil ondiions, Thn, fom fundmnl s of soluions o h gin diffnil quion. 9
Exmpl 6: Appl Thom..5 of Find h fundmnl s spifid b Thom..5 fo h diffnil quion nd iniil poin, In Sion., w found wo soluions of his quion:, Th Wonskin of hs soluions is W, = - so h fom fundmnl s of soluions. Bu hs wo soluions do no sisf h iniil ondiions sd in Thom..5, nd hus h do no fom h fundmnl s of soluions mniond in h hom. L L nd b h fundmnl soluions of fthm..5., ;,
Exmpl 6: Gnl Soluion of Sin nd fom fundmnl s of soluions,,, d d, Soling h quion, w obin, osh, sinh Th Wonskin of nd is W osh sinh osh sinh sinh osh Thus, fom h fundmnl s of soluions indid in Thom..5, wih gnl soluion in his s k osh k sinh 5
Exmpl 6: Mn Fundmnl Soluion Ss of Thus S, S osh,sinh, boh fom fundmnl soluion ss o h diffnil quion nd iniil poin, In gnl, diffnil quion will h infinil mn diffn fundmnl soluion ss. Tpill, w pik h on h is mos onnin o usful.
Summ To find gnl soluion of h diffnil quion p q, w fis find wo soluions nd. Thn mk su h is poin in h inl suh h W,. I follows h nd fom fundmnl s of soluions o h quion, wih gnl soluion = +. If iniil ondiions psibd poin in h inl wh W, hn nd n b hosn o sisf hos ondiions.
.: Complx Roos of Chisi Equion Elmn Diffnil Equions nd Bound Vlu Poblms, 9 h diion, b Willim E. Bo nd Rihd C. DiPim, 9 b John Wil & Sons, In. Rll ou disussion of h quion b wh,, b nd onsns. Assuming n xponnil soln lds o hisi quion: b Qudi fomul o foing ilds wo soluions, & : b b If b <, hn omplx oos: = + i, = - i Thus i i,
Eul s Fomul; Complx Vlud Soluions Subsiuing i ino Tlo sis fo w obin Eul s Subsiuing i ino Tlo sis fo, w obin Eul s fomul: i i i n n n n n i sin os Gnlizing Eul s fomul, w obin n n n n n n!!! Thn i i sin os i i Thfo i i i i sin os sin os i i i i i i i sin os sin os 5
Rl Vlud Soluions b Ou wo soluions hus f omplx-lud funions: os i sin os i sin 5 W would pf o h l-lud soluions, sin ou diffnil quion hs l offiins. To hi his, ll h lin ombinions of soluions hmsls soluions: os 6 i sin Ignoing onsns, w obin h wo soluions os, sin 7 6
Rl Vlud Soluions: Th Wonskin Thus w h h following l-lud funions: os, sin 8 Chking h Wonskin, w obin W os sin os sin sin os 9 Thus nd fom fundmnl soluion s fo ou ODE, nd h gnl soluion n b xpssd s b b b,, i os sin 7
i Exmpl of Consid h diffnil quion sin os Consid h diffnil quion Fo n xponnil soluion, h hisi quion is.5 9 p, q i i Thfo, sping h l nd imgin omponns, nd hus h gnl soluion is /, / nd hus h gnl soluion is sin os sin os / / / 8 8
Exmpl of Ui Using h gnl soluion jus dmind d / os sin W n dmin h piul soluion h sisfis h iniil ondiions nd ' 8 6 So / 8 Thus h soluion of his IVP is / os sin, 7 Th soluion is ding osillion / 5 os sin 6 8 9
Exmpl Consid h iniil lu poblm 6 8 5,, ' Thn 6 Thus h gnl soluion is And / Th soluion of h IVP is / os / Th soluion is displs gowing osillion 8 5 i sin / os, / sin / os / sin / 5 6 8 5
Exmpl Consid h quion 9 Thn 9 i Thfo, nd hus h gnl soluion is os sin Bus, h is no xponnil fo in h soluion, so h mpliud of h osillion mins onsn. Th figu shows h gph of wo pil soluions solid : os dshd : os sin / sin 6 8
.: Rpd Roos; Rduion of Od Elmn Diffnil Equions nd Bound Vlu Poblms, 9 h diion, b Willim E. Bo nd Rihd C. DiPim, 9 b John Wil & Sons, In. Rll ou nd od lin homognous ODE b wh,, b nd onsns. Assuming n xponnil soln lds o hisi quion: b Qudi fomul o foing ilds wo soluions, & : b b Whn b =, = = -b/, sin mhod onl gis on soluion: b /
Sond Soluion: Mulipling Fo W know h W know h Sin nd linl dpndn, w gnliz his soluion soluion p, g ppoh nd mulipl b funion, nd dmin ondiions fo whih is soluion: b b / / li Thn b b / / soluion b / b b b / / b b b b b b b / / / /
b Finding Mulipling Fo Subsiuing diis ino ODE w sk fomul fo : Subsiuing diis ino ODE, w sk fomul fo : / b b b b b b b b b b b b b b b b b b k k
Gnl Soluion To find ou gnl soluion, w h: k k b / k b / b / b / b / k k b / Thus h gnl soluion fo pd oos is b b b, = =-b/ b / b / 5
Wonskin Th gnl soluion is b / b / Thus soluion is lin ombinion of b / b /, Th Wonskin of h wo soluions is W b b, b / b / b / b / b / b / b fo ll b / b Thus nd fom fundmnl soluion s fo quion. 6
Exmpl of Consid h iniil lu poblm Consid h iniil lu poblm Assuming xponnil soln lds o hisi quion: g p q So on soluion is nd sond soluion is found: Subsiuing hs ino h diffnil quion nd simplifing ilds ' " k k k simplifing ilds wh bi onsns.,, k k k nd k k 7 7
Exmpl of Ling k k nd, nd So h gnl soluion is No h boh nd nd o s gdlss of h lus of nd Using iniil ii ondiions. nd ' 5.5, 5. 6 Thfo h soluion o h IVP is 5 7.5 5..5..5..5. 8
Exmpl of Consid h iniil lu poblm.5,, / Assuming gxponnil soln lds o hisi quion: q.5 Thus h gnl soluion is / / Using h iniil ondiions: / / / Thus, / / / 9
Exmpl of Suppos h h iniil ii slop in h pious poblm ws insd, Th soluion of his modifid d poblm is / / Noi h h offiin i of h sond 6 m is now posii. This mks big diffn in h gph, sin h xponnil funion is isd o posii pow: / d : / blu : / / 7 5