Question

I am not sure about the eigen vectors or the eigen values would like confirmation and the solutions for part B as well, Thank you.

(1 point) Consider the linear system = [3] -3 -2 5 3 y. -3-1 a. Find the eigenvalues and eigenvectors for the coefficient matrix. -3+1 di vi 5 -i and 12 13 5 b. Find the real-valued solution to the initial value problem โปร์ -3y1 - 2y2 5y1 + 3y2, yı(0) = -10, y2(0) = 15. Use t as the independent variable in your answers. (t) = y2(t) = Note: You can earn partial credit on this problem.

Answer 1

If you have any doubts in the solution please ask me in comments ....here i used basic concept of ode solution

-3 -2 gio) S9 sen ; g' = 3] 7 5 -3 -2 where y = ve Dy;] A = 92 5 3 ana 2,1t) Let Solution of System 0 is ü (t) = Cie N & Ge where 2,12 are eigen values of A m,, n2 are corrosponding eigen vectors of Aodz. ut a be eigen value of А 1A-XI) =0 -3-X -2 - (3-1)(3-1) +10 = 0 3-) a=t (1-3) (X+3) +10. =0 q² - g =-10 het dat - 3- then 52, +(3-4)X = 0 X,= (3+1) 5 3-i 1[*]=1:1 [:] [*1 } [ ****] Vi=11**] ña let a=- 5X +(3+i) x = 0 *[***]011): [ 1951] 3-i 5 22 ha = M₂ 3til AL V Answer so we have 5 22 3-i = -1 Jž Ansurer 5

solue IUP Now to - - 34, - 242 y, Y' 4,(0) = -10 92 (0)=15 sy, + 342 j wrote in system form A '' is By previous part solution of System (4) }(t) = cetton + Geazt na By O equation ^,,dz are eigen value of a M M are eigen vectors -3ti By previous paut, 9, = i, 22 = -4, 9,- where >> • įt) = geht [ 35'] + q est zeit [31] isine) ["35" ] [3] > g (+) = C, (Costt +2 ( cost -isint) -3 cost - sint & il cost - 3sint) +6 -3605 + - Sint til-cost #3sint 560st - ( 15 Sint) 5 cost + i(5 sint) - - 3 cost - sint Cost - 3 sint (6,+G) + (Cio-i (2) I 5 cost 5 sint -3 Cost - Sint č. Cost - 3sint 5(+) int] + 5 cost 5 sint Now 5o = [15] " C+₂= Z, C, d-Gi- By eq. (5) constant -3 [is] +6 [ ] 5 → By comparison we have @ @=3 = -1 -3 Cost - sint cost - - 3sint ỹlt) = 3 "] -1 ] 5 cost 5 sint

-10 cost » ū(t) = 15 cost - 5 sint. -10 cost, > 9,(t) = yz(t) = ] Answer. 15 wst - 5 sint.