Question
A. Find the most general real-valued solution to the linear system of differential equations z' =...
a. Find the most general real-valued solution to the linear system of differential equations z' = -6 -4. 1 -6 2. xi(t) = C1 + C2 22(t) S b. In the phase plane, this system is best described as a source / unstable node sink / stable node saddle center point / ellipses spiral source spiral sink none of these
Answers
solution @ Given system ។ x'= (3-** let Il 24 ( ) (3):1513 visz - 6 4 ] 4 34 - 61 A - 4 22 - 64 ار B and x 4 - 6x2 oso 44 H = (D+1)x4, 24. + ®+6)x2 = 0 Multiplying by (0+6) in 6 and adding @ - P + 6 x + (2 +6 x = ô (to) to 4 h adoling - o (@+6) ? x2 + 4x2 D +36 + 120 + 4] X2 + 12 D + 40) X2 Auxilliary equation ma +12m +40 5o - 12 2 I 1744-160 m - of-12 + -16 -6.2 zi. im = -12 £ 1(4) 2 Hence 67 X260) e [ G. Copet + Csinet) -6€ * Cacoszt teasinat} +2**(-29 sinat +26 Coszt] putting in 6 és (a cost + C sinat at] -at y it' (-29 sínat. + 2 Ce cos2+) =34 4 (t) =2ebt (-4 sinat + c cost] ot ge sinzit + G COS2) Now 34 (A) xift) EG (a cosat + casinat) com] X(t) = -6t -6t де Cosat 2 e sinat G t C xt) = il cat sinat est corat be characteristic equation is (A-JI) -6rd -6-) (-6-13 + 4 =(6+d)? +4= 36 + 12 + 12 444 = jet12+40 -12 d = 144 - 160 2 2 = - 12 € (4) d - 12 + 2i values are complex → option = 11 centrepoint ellipses
![solution @ Given system ។ x= (3-** let Il 24 ( ) (3):1513 visz - 6 4 ] 4 34 - 61 A - 4 22 - 64 ار B and x 4 - 6x2 oso 44 H =](http://img.homeworklib.com/questions/91700b10-10fb-11eb-997d-5dc4dddb5e36.png?x-oss-process=image/resize,w_560)
![-12 + -16 -6.2 zi. im = -12 £ 1(4) 2 Hence 67 X260) e [ G. Copet + Csinet) -6€ * Cacoszt teasinat} +2**(-29 sinat +26 Coszt]](http://img.homeworklib.com/questions/92b3b030-10fb-11eb-a042-f1c20d661125.png?x-oss-process=image/resize,w_560)
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