Question

Solve the initial value problem with 4 x'(t) = A, fort > O 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. x(o)= [1] A-[18 -16] -2 - 4 10 -16 2 -120 1 a. x(t)= (0,0) is a saddle point 5 2 120 b. x(t)= 1 + 6 le -61 (0,0) is an attractor 5 C. x(t)= o[1] - [3 ].-121+6[71].-*, (0,0) is an attractor - [3](1++[1].". (0,0) is repeller - [3 ]-21+[1]- d. x(t)= e. *(t)= - -61 (0,0) is a saddle point

Answer 1

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AO-2-4 x'ct's -Y -16 The eigen values for the metrie A 1 A-911=0 -2-2-4 1076-;? at 182 + 72 = 0 1=-6, -12 find the eigen vectors 1,5-6 [="][":]-[:] uv, - UU₂ 20 uv, = Uvq , = U2 can take yz he then ziil 2-12 10 -4 ul():[:] 10 -4 100,- UV, :0 10V, - UV2 let U2=1 the wa 0,2 2/3 m, (*) The general solution i given by at ๆ ๆ X(t)= Gent Ce

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