Question

2 Y, Y(t) 2 dY 5. For the system dt - 2. a) Write the general solution. b) State if the origin is a spiral sink, or a source, or a center. c) Write the natural period and the natural frequency of the solutions. d) Do the solutions go clockwise or anti clockwise around the origin? (0) e) Write the particular solution that corresponds to ly(0) =

Answer 1

Solvention, y'= []Y compare with Y'AY. To find eigenvalue consider LA-HI] =0 12 -d 2 - 2-d (2-d) (2-d) +2=0 4-20-20 +2+2=0 12²42 +6=0 threacteristic Roots are. -bt v62_496 a za a=1; b=-4,(=6 d = 4 + √16-24 d=41 vsi d=4Izrá W = 2 2 5 il

Eigenvalues cre. di- 2tr dz=2-ed aga (di = 2 +158] [de=z-vodi Eigen vectors consider [A-dI] K = 0 for Td = a trei R₂ → Rz tri li rilin7-10 LO 1677-407 y is free vociable - put lyst pe= -Fit put t = !. eigenvecter és C-24, 1)

for d = 2-Fi [A-dI]=0 [edit] (!)=(8) R2- RztR, Rea Petk, lo ] (376) free veriable 4 is g=t, ze=rzit ter pot t=1 Eigen vector is (vad, 1) General sein. y con = e, $215.30-krat ] + ca este un
i
© If we compare eigen velue. d=2 I rze with dadaßi we get 2=2 , B=r =) (270) - the solution spiεal away the origin =) origin is 'spital sovece. Natuɛal period=3 from 27 L-12T Natioecal teequeary a choose a vector 2][:]=] : recter v=147 lie in fiest quadeant =) we haweanklockcise &piecel around - ocigin. -) we have anticlockwise spital. aεound origin.

WWWB ERNARD © Cen= (4) you lovely) - podaril. *ip 1 Juel-** (63"] inco rc, 1-82d79 al 1200) 7 lyco)) TIF 31 J = c, ( - 1 C[ .0]=+ =) -vz.de +41 = ). rzecz + C2 =-1 => 3 = Tekin Y = ithai (-3.] <2*3*+ isi [171 Jezuri .