Question

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(1 point) a. Find the most general real-valued solution to the linear system of differential equations t' = 1 -6 x1 (t) = C1 + C2 x2(t) b. In the phase plane, this system is best described as a O source / unstable node O sink / stable node O saddle O center point / ellipses Ospiral source O spiral sink O none of these

Answer 1

Ques 2!= 5* -;]? A= [] First, we will find eigenvalue of A:- -bord -4 -6-1 -6 tet land +40 =0 Sol d=-12 + |144-160 2 d= - 12 + 16 - 12 t Fib 2 2 N= -6+ gi so, elgen values are -6tai and -b-di So, we will find eigen vector for these eigen values:- For h = obtai and do =-6-2i For di=-6-2i V A-d, I - ziu si Av=do (A-d, I) 1-0 have with the help of Gaussian elimination: we homogeneous System So, we will solve 있애 Ro>Rilai . 1:] 0 V I ai I ai o

o بالا - 22. . Cosatt isinat Now, Rq » Rq -R, [6.13] So, sataisa So, 24 = 36, = -2ixa 112=362 So, the general solution is X= -2in Solution set is Xg Let X2=1, 6,= Similarly For da we have Ve = Now, = -642 So, 2=-6, ß = 2 Now, eigen vector associated with da is :- So, the solution will be: - edt Le Bi] paht (cos2f + i smøt) [] gicosat + 2; 2 sinat [-2] [si] - [si] et 6+2i)t e - e-bt

50) we can write two linerely indebendent solution- X(t)= e-bt C, 1-25142A Cosat tal glosat singt (b) Here the eigin values are -6 tai anel -6-2i. Thase eigen values are complex conjugale so, it is a spiral point which is asymptotically stable because the real kart of roots ng negal lahue. so, it is spiral point with which is asymptotically stable . ) It will be if D70 To, spiral sourie and To<4D cunstable) it where, TE ard a,d foom any green (1:1) [-] matices 501 15 for (6) امی T=-12 To Tro so, it is hot spiral sowile And De adobe so, D36+4= 40 Drol And та - 144 < 4.0 (soj stable)

S9 it is a spiral Х), 6 bе (у г<ә, 72, 73<чЯ (44)