Question

4) Given the system of equations {x-3x2 + 2x2 lxa a) Rewrite the system in the form of x' = Aš b) Solve for the general solution using eigenvalues and eigenvectors c) Sketch the eigenvectors and a few typical trajectories indicating direction of solutions. 8 6 2 -10 -8 6.4 -2 6 8 10 2 -4 -6 -10

Answer 1

4 m,15 - 12 a ) A 2 C24 al 02 ) b) Now Given system m of equation is nal=-30, +2412 The system is '= AY , where &' where you can Ma' & (32) 1A-101=0 es (1-2)~350 12-22-3=0 e) (1-3)(1+1)=( > 133,-1 eigen values are values are 3,1. so the critical point is ů so the eigen Here the system is linear

(00). so (CO) is saddle & unstable, since eigen values are real unequal & opposite signe For A X = 3X 12:30, => 3u1dU2=0 - 381 +242 = 342 > 30, +4220 Take 2=34 . u=-9 so the eigen verton in a GERTS For d=a, AX=-X GEIR 201 -U23 => M-42=0 -3*, t2M2 =-82. ) q; -12-0 Take Uz = e => U = 4 so the eigen vector is a Therefore the genera eneral solution th e) The phase diagrame is U₂ = U, (1) 142 u M20-30