Question

Solve the initial value problem with x'(t) = A, for t20 with x(0)= Classify the nature of the origin as an attractor, repeller, or saddle point of the dynamical system described Ax=b. Find the directions of greatest attraction and/or repulsion. -2 -4 A= 10 -16 127 a. X(t)= is a saddle point b. X(t)= 121 + 6 (0,0) is an attractor 1 --[1]26. (0,0) --[] $]e=121+6[71]e-, (0,0) is an attractor d. x(e) = - [] -e[1]26. (0,0) is repeller e. x(t)= [1]e-61, (0,0) is a saddle point C. x(t) = e 121 + 6 2 121+ 5

Answer 1

Solution hiven initial value problem -4 AS X'!+) - AXH), foro +70 with X70)= [1] where [ie -7] Eigenvalues of matrix A are det (A-NI) Adil ----4 -16-1 10 4+184+72 (+12) (476) -) di E-12, d = -6 Now , de-12 The eigen vector associated with (A-diI) = [***]-[53] [] Them 100 xp = 4X2 =) 5X1=9X2 So, we have that the eigenvector for darle is % | The eigenvector associated with dq = -6 (A-deI) = Then 4x, = 4X2 =) X1X2 So, we have that the eigin vector for de a-6 is *v-]

X10). [3] looking for constants ei, lg such that Civi+ GV₂ = xo) [v, Ve X103] - 1 60-6 1:31 ]-[:-]-( Rp3R, FR₂ R19R-SR Thus Goal and ca 6 so we have that 1 -but X(+) =- -124 + 6 S an Since the two eigenvalues are both negative, The origin is attractor (or, sink) of the dynamical system given by x'() - Axit) Answer e 12t +6 -but X6+)=- 10,0) is an attractor b. 5