Boyce/DiPrima 9 th ed, Ch 7.6: Complex Eigenvalues

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1 BocDPm 9 h d Ch 7.6: Compl Egvlus Elm Dffl Equos d Boud Vlu Poblms 9 h do b Wllm E. Boc d Rchd C. DPm 9 b Joh Wl & Sos Ic. W cosd g homogous ssm of fs od l quos wh cos l coffcs d hus h ssm c b w s ' A wh A

2 Cojug Egvlus d Egvcos W ow h s soluo of ' A povdd s gvlu d s gvco of A. Th gvlus h oos of da-i d h cospodg gvcos ssf A-I. If A s l h h coffcs h poloml quo da-i l d hc compl gvlus mus occu cojug ps. Thus f λ µ s gvlu h so s λ - µ. Th cospodg gvcos cojugs lso. To s hs cll A d I hv l s d hc A I A I A I

3 Cojug Soluos I follows fom h pvous sld h h soluos cospodg o hs gvlus d gvcos cojugs cojugs s wll sc

4 Empl : Dco Fld of 7 Cosd h homogous quo ' A blow. A dco fld fo hs ssm s gv blow. A dco fld fo hs ssm s gv blow. Subsug fo d wg ssm s A-I w ob

5 Empl : Compl Egvlus of 7 W dm b solvg da-i. Now 5 Thus ± 5 ± ± Thfo h gvlus - d - -.

6 Empl : Fs Egvco of 7 Egvco fo - : Solv I A b ow ducg h ugmd m: Thus choos

7 Empl : Scod Egvco of 7 Egvco fo - - : Solv I A b ow ducg h ugmd m: Thus choos

8 Empl : Gl Soluo 5 of 7 Th cospodg soluos of ' A s s cos s cos u Th Wos of hs wo soluos s Thus u d v l-vlud fudml soluos of ' A wh gl soluo c u c v. cos s cos s v [ ] cos s s cos W

9 Empl : Phs Pl 6 of 7 Gv blow s h phs pl plo fo soluos wh cos c c s s cos Ech soluo jco ppochs og log spl ph s sc coods poducs of dcg pol d s o cos fcos. Th gph of u psss hough sc u. Smll h gph of v psss hough. Th og s spl po d s smpocll sbl.

10 Empl : Tm Plos 7 of 7 Th gl soluo s c u c v: As lv o phs pl plos w c gph o c c c c cos s s cos s fuco of. A fw plos of gv blow ch o dcg oscllo s.

11 Gl Soluo To summz suppos λ µ λ - µ d h ll l d dsc gvlus of A. L h cospodg gvcos b b b Th h gl soluo of ' A s wh c u cv c c λ cos µ bs µ v s µ cos µ λ u b

12 Rl-Vlud Soluos Thus fo compl cojug gvlus d h cospodg soluos d cojugs lso. To ob l-vlud soluos us l d mg ps of h o. To s hs l b. Th wh λ λ µ λ b cos µ s µ λ cos µ bs µ s µ b cos µ u v λ cos µ bs µ v s µ b cos µ λ u l vlud soluos of ' A d c b show o b ll dpd.

13 Spl Pos Cs Egvlus d Tjcos I pvous mpl gl soluo ws cos c c s s cos Th og ws spl po d ws smpocll sbl. If l p of compl gvlus s posv h jcos spl w uboudd fom og d hc og would b usbl spl po. If l p of compl gvlus s zo h jcos ccl og h ppochg o dpg. Th og s clld c d s sbl bu o smpocll sbl. Tjcos podc m. Th dco of jco moo dpds o s A.

14 Empl : Scod Od Ssm wh Pm of Th ssm ' A blow cos pm α. Subsug fo d wg ssm s α A-I w ob N solv fo ms of α: α 6 ± α α α α α

15 Empl : Egvlu Alss of α ± α 6 Th gvlus gv b h qudc fomul bov. Fo α < - boh gvlus l d gv d hc og s smpocll sbl od. Fo α > boh gvlus l d posv d hc h og s usbl od. Fo - < α < gvlus compl wh gv l p d hc og s smpocll sbl spl po. Fo < α < gvlus compl wh posv l p d h og s usbl spl po. Fo α gvlus pul mg og s c. Tjcos closd cuvs bou og & podc. Fo α ± gvlus l & qul og s od Ch 7.8

16 Scod Od Soluo Bhvo d Egvlus: Th M Css Fo scod od ssms h h m css : Egvlus l d hv oppos sgs; s sddl po. Egvlus l dsc d hv sm sg; s od. Egvlus compl wh ozo l p; spl po. Oh possbls s d occu s sos bw wo of h css lsd bov: A zo gvlu occus dug so bw sddl po d od. Rl d qul gvlus occu dug so bw ods d spl pos. Pul mg gvlus occu dug so bw smpocll sbl d usbl spl pos. b ± b c

17 Empl : Mulpl Spg-Mss Ssm of 6 Th quos fo h ssm of wo msss d h spgs dscussd Sco 7. ssumg o l focs c b pssd s: d d m d m Gv h quos bcom ' d ' wh ' d ' o d m m d m d m 5 d 9 m m ' d ' ' '

18 ' ' ' d ' Empl : Mulpl Spg-Mss Ssm of 6 Wg h ssm of quos m fom: ' Assumg soluo of h fom wh mus b gvlu of h m A d s h cospodg gvco h chcsc poloml of A s 5 ldg h gvlus: A d

19 Empl : Mulpl Spg-Mss Ssm of 6 Fo h gvlus h cospodg gvcos A A A A ' ' ' ' d 8 6 d 8 6 Th poducs ld h compl-vlud soluos: 8 8 d 8 cos 6 cos s s 8 s 6 s cos cos s cos 8 6 cos cos s s s s cos cos s cos v u v u

20 Empl : Mulpl Spg-Mss Ssm of 6 Af vldg h u v u v ll dpd h gl soluo of h ssm of quos c b w s wh c c c c ' ' ' d ' cos s cos s cos s cos s c c c c s cos 6s 6cos s cos 8s 8cos b coss. Ech soluo wll b podc wh pod π so ch jco s closd cuv. Th fs wo ms of h soluo dscb moos wh fquc d pod π whl h scod wo ms dscb moos wh fquc d pod π. Th moos of h wo msss wll b dff lv o o oh fo soluos volvg ol h fs wo ms o h scod wo ms.

21 d ps h moo of h msss d ' ' Empl : Mulpl Spg-Mss Ssm 5 of 6 To ob h fudml mod of vbo wh fquc c c occus wh d To ob h fudml mod of vbo wh fquc c c occus wh d Plos of d d pmc plos show fo slcd soluo wh fquc u HL HL Plos of h soluos s fucos of m Phs pl plos

22 Empl : Mulpl Spg-Mss Ssm 6 of 6 Plos of d d pmc plos show fo slcd soluo wh fquc u HL HL - - Plos of h soluos s fucos of m d ps h moo of h msss d ' ' 6 8 Phs pl plos Plos of d pmc plos show fo slcd d soluo wh md fqucs ssfg h l codo sd -5 HL - Plos of h soluos s fucos of m Phs pl plos

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