Question

(1 point) 0 -5 a. Find the most general real-valued solution to the linear system of differential equations z' = 1. сл xi(t) + C2 x2(t) b. In the phase plane, this system is best described as a source / unstable node Osink / stable node Osaddle O center point / ellipses spiral source Ospiral sink O none of these

Answer 1

first calculate eign value then eign vector by corresponding eign value further put value into general form solition. In part be both eign value are pure imaginary means real lamda is zero so phase is center Or ellipses.
Page 1 -S no 15 Lend most general real value problem Solution Arswer porta first find ein value of matrix A= know characteristic equation IA-AIlo where iis ein volio o coe 1829-07- - 2 EX 4+25 =0 → Tetsi V whore i=F1 (cmagn oggy ourbes) cion ralue are si ond -si to lesi feod. eien recher according {A-1I) () = Sleppose O 04 a2 93 eim vector (A-11)0 al D x=lor then si taken -si -s 1) )(*)=40 -si S 592/-si zazi aya csiai -S42 20 = saursia2=0 Put do this sian=0. sazi D. (si-si) (0)=0 o this is zero any constant e ci Ghen

page2 frist eiga recher v Lizsi) = 0914 most general vi Cassi); Sendanly for x=-si use is equation n(=s) = (1) -si 0 (t ::]) %) -si -azi sicu-592 20 7 Sau + si92=0 put here SX-azi + sian=0. constant c' 92 my (S-)a) =o. then 04 = -ci $0 einn veckor v»(A==5) = 1.) [gmoval from) of gren Systems -sit e generale solution PC]= c, Vịch). of Carz edzt Y, LG ILLU =(9) L sit +[C] 57 1 is conslenken 31 coherre and 2 Penale mucoes sit e-sit it c2 J. LE Jele

Ryy | Page 3 29ben cion values prey cmgency means Real lo then in phase plane system is like this called center ay point or ellipses = cestor point ellipses gloes