BoDiPrima 9 h d Ch 7.9: Nohomogou Liar Sm Elmar Diffrial Equaio ad Boudar Valu Prolm 9 h diio William E. Bo ad Rihard C. DiPrima 9 Joh Wil & So I. Th gral hor of a ohomogou m of quaio g g aralll ha of a igl h ordr liar quaio. Thi m a wri a ' P g whr g g g g P g
Gral Soluio Th gral oluio of ' P g o I: α < < β ha h form whr v i h gral oluio of h homogou m ' P ad v i a ariular oluio of h ohomogou m ' P g.
Diagoalizaio Suo ' A g whr A i a diagoalizal oa mari. L T h oigular raform mari who olum ar h igvor of A ad D h diagoal mari who diagoal ri ar h orrodig igvalu of A. Suo aifi ' A l dfid T. Suiuig T io ' A w oai T' AT g. or ' T - AT T - g or ' D h whr h T - g. No ha if w a olv diagoal m ' D h for h T i a oluio o h origial m.
Solvig Diagoal Sm Now ' D h i a diagoal m of h form h h r r h r h r whr r r ar h igvalu of A. Thu ' D h i a uould m of liar fir ordr quaio i h uow whih a iolad ad olvd aral uig mhod of Sio.: h r h r h r d h r r r
Solvig Origial Sm Th oluio o ' D h ha omo r r r h d For hi oluio vor h oluio o h origial m ' A g i h T. Rall ha T i h oigular raform mari who olum ar h igvor of A. Thu wh mulilid T h od rm o righ id of rodu gral oluio of homogou quaio whil h igral rm of rodu a ariular oluio of ohomogou m.
Eaml : Gral Soluio of Homogou Ca of 5 Coidr h ohomogou m ' A g low. No: A i a Hrmiia mari i i i ral ad mmri. g A No: A i a Hrmiia mari i i i ral ad mmri. Th igvalu of A ar r - ad r - wih orrodig igvor Th gral oluio of h homogou m i h ξ ξ
Eaml : Traformaio Mari of 5 Coidr h raformaio mari T of igvor. Uig a Sio 7.7 omm ad A Hrmiia w hav T - T * T T rovidd w ormaliz ξ ad ξ o ha ξ ξ ad ξ ξ. Thu ormaliz a follow: Th for hi hoi of igvor ξ ξ T T
Eaml : Diagoal Sm ad i Soluio of 5 Udr h raformaio T w oai h diagoal m ' D T - g: Th uig mhod of Sio. 9
Eaml : Traform Ba o Origial Sm 4 of 5 W u h raformaio T o oai h oluio o h origial m ' A g: 6 5 4
Eaml : Soluio of Origial Sm 5 of 5 Simlifig furhr h oluio a wri a 5 4 No ha h fir wo rm o righ id form h gral oluio o homogou m whil h rmaiig rm ar a ariular oluio o ohomogou m. 5 4
Nodiagoal Ca If A ao diagoalizd rad igvalu ad a horag of igvor h i a raformd o i Jorda form J whih i arl diagoal. I hi a h diffrial quaio ar o oall uould au om row of J hav wo ozro ri: a igvalu i diagoal oiio ad a i adja oiio o h righ of diagoal oiio. Howvr h quaio for a ill olvd ouivl arig wih. Th h oluio o origial m a foud uig T.
Udrmid Coffii A od wa of olvig ' P g i h mhod of udrmid offii. Aum P i a oa mari ad ha h omo of g ar olomial oial or iuoidal fuio or um or rodu of h. Th rodur for hooig h form of oluio i uuall dirl aalogou o ha giv i Sio.6. Th mai diffr ari wh g ha h form u λ whr λ i a iml igvalu of P. I hi a g mah oluio form of homogou m ' P ad a a rul i i ar o a ohomogou oluio o of h form a λ λ. Thi form diffr from h Sio.6 aalog a λ.
Eaml : Udrmid Coffii of 5 Coidr agai h ohomogou m ' A g: Aum a ariular oluio of h form Aum a ariular oluio of h form whr h vor offi a d ar o drmid. Si r - i a igvalu of A i i ar o ilud oh a - ad - a miod o h rviou lid. d a v
Eaml : Mari Equaio for Coffii of 5 Suiuig i for i our ohomogou m ' A g d a v w oai Equaig offii w olud ha Ad A a A a Aa Ad A A Aa a a
Eaml : Solvig Mari Equaio for a of 5 Our mari quaio for h offii ar: Aa a A a A Ad From h fir quaio w ha a i a igvor of A orrodig o igvalu r - ad h ha h form α a α W will o h lid ha α ad h a
Eaml : Solvig Mari Equaio for 4 of 5 Our mari quaio for h offii ar: Suiuig a T αα io od quaio Ad A a A a Aa Thu α ad olvig for w oai α α α α α α α α A hoo
Eaml : Pariular Soluio 5 of 5 Our mari quaio for h offii ar: Solvig hird quaio for ad h fourh quaio for d i i raighforward o oai T d T -4-5. Ad A a A a Aa i i raighforward o oai T d T -4-5. Thu our ariular oluio of ' A g i Comarig hi o h rul oaid i Eaml w ha oh ariular oluio would h am if w had ho ½ for o rviou lid iad of. 5 4 v
Variaio of Paramr: Prlimiari A mor gral wa of olvig ' P g i h mhod of variaio of aramr. Aum P ad g ar oiuou o α < < β ad l Ψ a fudamal mari for h homogou m. Rall ha h olum of Ψ ar liarl idd oluio of ' P ad h Ψ i ivril o h irval α < < β ad alo Ψ' PΨ. N rall ha h oluio of h homogou m a rd a Ψ. Aalogou o Sio.7 aum h ariular oluio of h ohomogou m ha h form Ψu whr u i a vor o foud.
Variaio of Paramr: Soluio W aum a ariular oluio of h form Ψu. Suiuig hi io ' P g w oai Ψ'u Ψu' PΨu g Si Ψ' PΨ h aov quaio imlifi o u' Ψ - g Thu u Ψ g d whr h vor i a arirar oa of igraio. Th gral oluio o ' P g i hrfor Ψ Ψ Ψ g d α β arirar
Variaio of Paramr: Iiial Valu Prolm For a iiial valu rolm ' P g h gral oluio o ' P g i d g Ψ Ψ Ψ Ψ Alraivl rall ha h fudamal mari Φ aifi Φ I ad h h gral oluio i I rai i ma air o row rdu mari ad olv ar quaio ha o omu Ψ - ad uiu io quaio. S aml. d g Ψ Ψ Ψ Ψ d g Ψ Φ Φ
Eaml : Variaio of Paramr of Coidr agai h ohomogou m ' A g: W hav rvioul foud gral oluio o homogou W hav rvioul foud gral oluio o homogou a wih orrodig fudamal mari: Uig variaio of aramr mhod our oluio i giv Ψu whr u aifi Ψu' g or Ψ u u
Eaml : Solvig for u of Solvig Ψu' g row rduio I follow ha u u 6 u u u
Eaml : Solvig for of Now Ψu ad h w mulil o oai afr ollig rm ad imlifig 6 o oai afr ollig rm ad imlifig No ha hi i h am oluio a i Eaml. 5 4
Lala Traform Th Lala raform a ud o olv m of quaio. Hr h raform of a vor i h vor of omo raform dod X: { } { } L L L X Th dig Thorm 6.. w oai { } { } L L L X { } X L
Eaml 4: Lala Traform of 5 Coidr agai h ohomogou m ' A g: Taig h Lala raform of ah rm w oai Taig h Lala raform of ah rm w oai whr G i h raform of g ad i giv G G AX X
Eaml 4: Trafr Mari of 5 Our raformd quaio i X AX G If w a h h aov quaio om X AX G or I A X G Solvig for X w oai X I A G Th mari I A - i alld h rafr mari.
Eaml 4: Fidig Trafr Mari of 5 Th Solvig for I A - w oai A I A Solvig for I A w oai A I
Eaml 4: Trafr Mari 4 of 5 N X I A - G ad h X or X
Eaml 4: Trafr Mari 5 of 5 Thu X To olv for L - {X} u arial fraio aio of oh omo of X ad h Tal 6.. o oai: Si w aumd hi oluio diffr lighl from h rviou ariular oluio. 5 4
Summar of Th mhod of udrmid offii rquir o igraio u i limid i o ad ma ivolv vral of algrai quaio. Diagoalizaio rquir fidig ivr of raformaio mari ad olvig uould fir ordr liar quaio. Wh offii mari i Hrmiia h ivr of raformaio mari a foud wihou alulaio whih i vr hlful for larg m. Th Lala raform mhod ivolv mari ivrio mari muliliaio ad ivr raform. Thi mhod i ariularl uful for rolm wih dioiuou or imuliv forig fuio.
Summar of Variaio of aramr i h mo gral mhod u i ivolv olvig liar algrai quaio wih varial offii igraio ad mari muliliaio ad h ma h mo omuaioall omliad mhod. For ma mall m wih oa offii all of h mhod wor wll ad hr ma lil rao o l o ovr aohr.