Question

(1 point) a. Find the most general real-valued solution to the linear system of differential equations ã' = 21(t) = C1 + c 22(t) b. In the phase plane, this system is best described as a source / unstable node sink / stable node Osaddle O center point / ellipses spiral source Ospiral sink O none of these

Answer 1

solution: Consider the system of differential equation x = = AR where is 2x2 matrix Then general solution to this system of differential equation is ² (t) = C, edota (1) + Czetzt n (2) where d, and de eigenvalues of A and no and their corresponding eigenvectors. were 2 (2) are Given 2 3 Here - 2 A = -4 We Firest find eigenvalues of A. Characteristic polynomial of A CA(d) = det (A-dI) I-d - -2 1 3 -4-d =(1-d) (-4-d) + 6 = (d-1) (d+4) +6 = d+ 4d-d-4+6 d²+3d +2 CA(d)=0 eigenvalues , put d²+ 3+2=0 d² +2dtd +2=0 d(d+2)+ 1 (2+2) = 0 (+1) (+2) = 0 d=-1, -2

3 Eigenvalues of A ace di =-1, d₂ = -2 di=-1 Eigenvector corresponding to (A-d, I n = 0 2 ECO 3 -3 m2 R. RL R₂ R2 3 C R₂ R₂ - Ri O 2₂ - free variable Let n=t n - 2₂=0 = n = ₂ = n = t - t n t + (1) Therefore, eigenvector corresponding to d, = -1 18 30 = Eigenvector corresponding to (A-da I) i = 0 d₂ = -2 3 3 -2 m2 R₂ → R₂ - Ri 3 3) nz free variable h = t let

in 30 - 22₂ = 0 =) no 2 na - ) n = 2+ 3 mi رابر) 좋t 7 3 ty - z Therefore, eigenvector corresponding to d2=-2 is n (2) dlm - Then , the given linear system general solution to differential eqreation is X, (t) 2 (t) The the C, et こ * + C₂ e 2t Gerli X2(+) = x, (t C e-t -2+ C₂ | 3 6 + X2 ( t) e-t e-t

of A 8 P & a are Non Linear y di, da are eigen value Linear deldet P a a Node , Saddle ) Stability Spiral, Centre didzER A dioch> o Node di so, da >0, unstable dico, d₂8o, asy. stable y diodeco Saddle Unstable di=dz = dER 5 d > 0 Node Unstable Node or Spiral a dko Node Asy. stable dizatiß, dz=d-iß a0 Spioral Unstable - ako Spiral Asy stable di=iß, dz=eiß Centre stable bed not asy stable Centre or Spiral x

b) Since di =-1 g d=-2 du dz E R dio d₂ = (-1) (-2)=2 >o and dico dzco in system is stable node