Boy/DiPrima 9 h d Ch 7.8: Rpad Eignvalus Elmnary Diffrnial Equaions and Boundary Valu Problms 9 h diion by William E. Boy and Rihard C. DiPrima 9 by John Wily & Sons In. W onsidr again a homognous sysm of n firs ordr linar quaions wih onsan ral offiins ' A. If h ignvalus r r n of A ar ral and diffrn hn hr ar n linarly indpndn ignvors n and n linarly indpndn soluions of h form r n n If som of h ignvalus r r n ar rpad hn hr may no b n orrsponding linarly indpndn soluions of h abov form. In his as w will sk addiional soluions ha ar produs of polynomials and ponnial funions. r n
Eampl : Eignvalus of W nd o find h ignvors for h mari: Th ignvalus r and ignvors saisfy h quaio 3 A A ri or To drmin r solv da-ri : Thus r and r. 4 4 3 3 r r r r r r r 3 r r
Eampl : Eignvors of To find h ignvors w solv by row rduing h augmnd mari: 3 A ri by row rduing h augmnd mari: Thus hr is only on linarly indpndn ignvor for h rpad ignvalu r. hoos
Eampl : Dirion Fild of Considr h homognous quaion ' A blow. 3 A dirion fild for his sysm is givn blow. Subsiuing r in for whr r is A s ignvalu and is is orrsponding ignvor h prvious ampl showd h isn of only on ignvalu r wih on ignvor:
Eampl : Firs Soluion; and Sond Soluion Firs Amp of Th orrsponding soluion r of ' A is Sin hr is no sond soluion of h form r w nd o ry a diffrn form. Basd on mhods for sond ordr linar quaions in Ch 3.5 w firs ry. Subsiuing ino ' A w obain or A A
Eampl : Sond Soluion Sond Amp 3 of From h prvious slid w hav A In ordr for his quaion o b saisfid for all i is nssary for h offiins of and o boh b zro. From h rm w s ha and hn hr is no nonzro soluion of h form. Sin and appar in h abov quaion w n onsidr a soluion of h form η
Eampl : Sond Soluion and is Dfining Mari Equaions 4 of Subsiuing η ino ' A w obain or η A η η A Aη Equaing offiins yilds A and Aη η or A I and A I η Th firs quaion is saisfid if is an ignvor of A orrsponding o h ignvalu r. Thus
Eampl : Solving for Sond Soluion 5 of Rall ha Thus o solv A Iη for η w row rdu h 3 A orrsponding augmnd mari: k η η η η η η
Eampl : Sond Soluion 6 of Our sond soluion η is now Ralling ha h firs soluion was k w s ha our sond soluion is simply sin h las rm of hird rm of is a mulipl of.
Eampl : Gnral Soluion 7 of Th wo soluions of ' A ar Th Wronskian of hs wo soluions is Thus and ar fundamnal soluions and h gnral soluion of ' A is [ ] 4 W
Eampl : Phas Plan 8 of Th gnral soluion is Thus is unboundd as and as -. Furhr i an b shown ha as - asympoi o h lin - drmind by h firs ignvor. Similarly as is asympoi o a lin paralll o -.
Eampl : Phas Plan 9 of Th origin is an impropr nod and is unsabl. S graph. Th parn of rajoris is ypial for wo rpad ignvalus wih only on ignvor. If h ignvalus ar ngaiv hn h rajoris ar similar bu ar ravrsd in h inward dirion. In his as h origin is an asympoially sabl impropr nod.
Eampl : Tim Plos for Gnral Soluion of Tim plos for ar givn blow whr w no ha h gnral soluion an b wrin as follows.
Gnral Cas for Doubl Eignvalus Suppos h sysm ' A has a doubl ignvalu r ρ and a singl orrsponding ignvor. Th firs soluion is ρ whr saisfis A-ρI. As in Eampl h sond soluion has h form ρ ρ η whr is as abov and η saisfis A-ρIη. Sin ρ is an ignvalu da-ρi and A-ρIη b dos no hav a soluion for all b. Howvr i an b shown ha A-ρIη always has a soluion. Th vor η is alld a gnralizd ignvor.
Eampl Ension: Fundamnal Mari Ψ of Rall ha a fundamnal mari Ψ for ' A has linarly indpndn soluion for is olumns. In Eampl our sysm ' A was and h wo soluions w found wr Thus h orrsponding fundamnal mari is Ψ 3
Eampl Ension: Fundamnal Mari Φ of Th fundamnal mari Φ ha saisfis Φ I an b found using Φ ΨΨ - whr Ψ Ψ whr Ψ - is found as follows: Thus Φ
Jordan Forms If A is n n wih n linarly indpndn ignvors hn A an b diagonalizd using a similariy ransform T - AT D. Th ransform mari T onsisd of ignvors of A and h diagonal nris of D onsisd of h ignvalus of A. In h as of rpad ignvalus and fwr han n linarly indpndn ignvors A an b ransformd ino a narly diagonal mari J alld h Jordan form of A wih T - AT J.
Eampl Ension: Transform Mari of In Eampl our sysm ' A was wih ignvalus r and r and ignvors 3 Choosing k h ransform mari T formd from h wo ignvors and η is k η T
Eampl Ension: Jordan Form of Th Jordan form J of A is dfind by T - AT J. Now and hn T T and hn No ha h ignvalus of A r and r ar on h main diagonal of J and ha hr is a dirly abov h sond ignvalu. This parn is ypial of Jordan forms. 3 T AT J