Question

(1 point) a. Find the most general real-valued solution to the linear system of differential equations a' 2.(0) z(t) C + c b. In the phase plane, this system is best described as a O source / unstable node sink/stable node O saddle O center point/ ellipses spiral source Ospiral sink

Answer 1

SOLUTIONS T = 3 2 - 3 Eigen value legevation for coefficient mahoy -3-1 -4 13 -1) = 0 -3-1 (-3-2) ²+4=0 +² +67+13=0 x=-6 ± √36-52 = -3 +21 2 Now eigen vector for , =-3-2i - 13723687] [2.3-03 J] [*]=[3] 2 zi R₂ R2 + Iiri (3][]= [:] choose X2=1 as free variable Hence zix - 4X₂ x = ² 2i =-zi i +12 Hence х + [2]

Now eigen vector for = -3+2i -3+3-2i [** -3 +3-21) „J[3)-(3) (*)-(3) -2i [i -zi R2 R2 - £R, [ 3 ][*)=(:) x2 is a free variable choose x2=1 -zix, - 442=0 x,= zi Hence [:] -

Hence its solution x, (t) = e ē3t [cos2t+isinet] +7(3) ē zi cosat - asinat i sinatt Cosat [ [ -> - 3t - 2 sinat 2 coszt ti ezt Coszt sinat Hence complete solutions :-2 et sinat x (t)= 9 2e +7+6L est cosat eßt simt 3 cosat Since Real part of eigen value is negative the trajectories will spiral csink) into the origin and in this the equilibrium solution will be asymtotically stable. case