Question

(i) Find the general solution by using eigenvalues and eigenvectors (ii) State whether the origin is stable, asymptotically stable, or unstable spiral (iii) State whether the origin is a node, saddle, center, or -1 2 2 -1

Answer 1

Soluton The given dfferenta al eqh i's = Ax 2 Here A = a2-Groce AA +(detA) (4-) 12-42+3= 0 A- Characlertsht e?" 3) 0 (posihive) > =し3 Corresponding ergen ve ct )-() X1 2-1 2-1 corresjoonding ve ctr ergen )-Ce) 2-3 X2 2-3 メ2ニーメ」 Hence enenal Solutron 3t e set+t Get C2e 3t - e Ans

2 AX. qulonum p0int of x- Origin 1,3 are eigene value of A Since wtuch are pasihive unstable . evey solutven i's Hence Solution 3t t C X both eigen values of A are positive of2ame rgn) ( 1& 3 Since Henee o, 0) s a NODE