Question

1) Find the general solution of di = Ay where Then sketch the phase portrait in the x-y plane, where Finally, classify the equilibrium solution at the origin as a source, spiral sink, etc. 2) Repeat for the matrix | 3 -31 -2 -2] 3) Repeat for the matrix 4 — 4) Repeat for the matrix [95 -9 15 but you don't need to sketch the phase portrait.

Answer 1

) he hare dg syglem diffrentul equolions Ay dt 3 -Qo Where - 15 EigervalucsmA are en A-Ar) -0 3 -I5-A 6 20+A) 5+A) -6 30085A + d-6 + २१५ d35A di-4, , = - | b he eieee corepnh td A c -6 3 -1 On ingal T»u poration om +Nr) 3 -1

Phoefor Ci- ニ0 C, = Ihen Let C i ree vaina&le eigenedtor Coeaprely to dr-y 1 Let be the eigenedo Cy to eigenualu C) A+ 0 3 1 de otainrof mar weget Poyfarm peratiens t 3 C On hiyuget - 3

C E -14t tB Det Ce -3 1

Same sgn wh d, dare negial Snce eas than 3ero Sinhaat the origin then equuluton 0s -3 A = let poceeding same A ae given byA-AI) Eyenvubues ofmutni , d3 eijemetos ar and 1 Coy () ga (C) which are

3t are two T-8lutis and and hene geneu lutiyt) d Ce BDeat ut e -3t -3 distine Ond opposite sign read Pmieegenales ve Seelalesat the or gin) eq uh labrimutin (3) Let A Cyrmawlus the Camapu genetue (A-AiI)C=0 are - are guen by Qnd Where Cand D ane eigenveetes Capaci h eigenuls Onsaloiy jet Ond D ahich ar

- 6i't and arc lutima the geneal pltii u And Hene t Desit Cebit dil3 -6i t ra past is zero SIRcE 1envalu oriny Ahexefor euilbim lution as Center h table at the oigin [: 5 det A= I5 Bpceediy ome aa in past genvalues ae A, = 12-6 A-I C0 egervaeha Cmaprady t eienali Cohese ase = 12+ 6i d 2-6 and d YCp

For d=12-6i 5 -3+ Ae 3+ 6i mef gmati ofter opphyiy operat we Jt Lobich is 1J. eijenued Smita'yud 12t 6e 3 ejenadaed1 6i 2+6i)t 7Cea)t Soletin4ymeutims -

Heno gencad luhin sgy ,13-6i)t glep d Ce (18-6i)t 1 Shie eigenali se Cale Cmpe but nof pus ly jimagmery oith in og mesy elpar gen than gess thegh uili luti Spial point th at the aigin intadle be hoviobr