Question

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(1 point) a. Find the most general real-valued solution to the linear system of differential equations ' = 1 x1 (1) = C1 + C2 x2 (1) b. In the phase plane, this system is best described as a source / unstable node O sink / stable node O saddle center point / ellipses O spiral source O spiral sink O none of these

Answer 1

solution: ca) *- []* o per me (* *]p] btw 21 (3) -6 first we find eigen value of - let a be an eigen value of tour then 1 IA-11-0 -6 d - 4 = 0 1 -6-d = + (-6-2) (-6-1) +u=0 = 16+2 7274=0 12+36 + 12-1+4-0-) 12+12+ + 40 =0 -12 t 144-160 Roots of ax²+bx+c=0 are x= b ± √6²40c 1 2a = -12+ F-160 -12+ 40 2 2 : -6+ ai let = -6tai, d2=-6-gi Eigen vects Corresponding di=-6+2i :- [A-1,11 ) = [:] [* - ] [*:)+ [:]

=) HF-22 de 24=di ui] xit Cgetzt, a) - diru -482=0 and 24 - 21x2=0. -) aixo - (911²x220 =) gix + 4x2 = 0 =) dixi=-482 =) ix = -2x2 x2 = x1= qixz Take X2=1 then lai = X, (say) -4 li dup-402=0, Ritgix2=0 - dont lail²x2= =) Qixi 422=0 » 14-822=0 ix=2x2 = x=2 - 2x = x= - ging Take 22=1 then x= -gi : eigen vecko to d2 = -a i : General solution Hilt) X2 .: eigen vector Corresponding to dia eigen vector Corresponding to th=-6-2i [A-d211 [x] = [o? » [34 a ] [ ] = [:] O Goresponding X2 (say) arbiterey constants こ 6 where ci and C2 are

24 (1) (top] сі е total ai] + ca 2-ai X2 (4) fai & staisty +c2 to-aut. -di 66-21) fot&ilt 210) = dig66+ (6+2) (-6-ai)t + (-ai) cg 66+ailt Ge he (t) = 6966-21) - 2017 aicié tl csatt isinat) - gicgect (cs (24) - i sin (au) Geot (osatt isinat) + Cg est (cosat - ismat). 7 Hg (t) , Lt) = diebt (963347 Giminate (268 2t + Cg isinat ) (616827+ tigsingt + Caosat - ila sinat). and Ag (t) =ębt 2014) = aiért -) [Cosat (4-62) + i sinat (a +62) ] and *g(t)= é bt [esat (Ct (2) + isinat (a-ca)]. let Cit Cg = k and C-Cq=d. kid av alo cibilcery Constanta then ni (t) = diebt (a cosattkisinat) and Yg (+) (k osat disinat) which is the required general solution ést +

(6) Result: Consider the linear system da at ax+by 3-c را) dy catdy dt of the form dizide and if both chasectecistic sols of ci) av then the critical point (0,0) of (1) ij asymstebily stable spiced diko here di=-6, tz = ai and di=-6co this system is best described as a stable spiral I spiral sink.
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