Question

X'=2 -5x, x(x) = -2 z LI -2 5

Answer 1

Ques: solve the System -5 |x, X (T) = [-2] -2 Ansi- The Given System X' (1) of the form A= Now So 2 -5 where (? - ] we find the solution the homogeneous System of quetion X'=AX first we find Eigen vector and eigenvector of A. Eigen value of A- [? - The characteristic polynomial IPI-A 1-0 19LO;B[7] = a-2 +5 0-1 at2 = (2-2) (2+2) +5=0 224 +5=0 => 78 +1 -0 (Ati) (2-1)=0 Then a = fi =0

di= Now & U2=1 so Eigenvalue of Ais and daai find Eigenvector Corresponding to hizi (A - l.I) u=0 { 13:21:15:) }6) - 609 ( 2-1 2-1] = (2-2) 4 - 5ų = 0 - cis + (-2-1) 4₂ = 0 Solve (i) & (ii) & (i), we get ly = (2+i) So eigen vector Corresponding to Ance is u=[] City Now we find Eigenvector Corresponding to Ag - - i (A - (-1). I) U=0 or (Atl. I) u=0 {[1-21 +1:} 10] -5 -2 ti (2 +i) 0, -50 20 + (-2+i) 02 =0 (1) Solve (ili) and (in we get U,= (2-1) = 11 2 ti U

"(21) Now Eigen vector Corresponding to do = -i, U- Now Solution X(t)= G M eart X(t) = G( 2 che + Gp4te trisint) +G to ve = [2 de licent risid) og Balkan (2 d Cart mesin) Xt) ati tesint) +G (2-1) (Cart Given X (T) = (-3 put 5 2 + X119=6) in @ , weget 1-3) = a[ dre) (CsT + i sian) 46 [4-e) (-ebih t) =41211) C1 + doo] +62 2-4 (-1-20) 13) -41371 -G (2+1) - Cg (2-1) ta (2+i) tG (2-1) = +2 (V) - G - G =75 (0) mulplity (2+i) in (uis & add with us, weget G (27) ta (2-1) = 2 -G (2 t) + (2+i) = 5l2te) G [2-1-2-i = 2+10+5i

C .-2i = 12+5 i 12+5 i -Qi Xi G 5 12 +12i -) 6= -5 +126 co 2 Put value of Ca in (uis weget -G+/- ( -5 +126)] = 5 5 + -5+12i 2 Solution -G= 10 -5 +121 [G=- [ 5+128 Rut value of G and g in 6 , weget required Tsung *14] (054 ) * ($121) 24 (Content - [54121)(2+i) (Cost tarint) + (-5124(0-1) (Cost-isid) (54122) (st + sint) + (-57121) (Cost-esit) Kontakt Pro Contestan - (51124) (Cost teisint) + (-54124) (Cost -as inut) 2 -291 +24294) Cost + (2-2,34 4 29294) o'sint (-5-42145 721 ) isht XL t): YCH I lois lae - asiat) - 5-121 -5 +121 Cost t 2 Cost +29 Cost

. tg ug e dat - t Co Une ant Hence solution of the system 2 Cost +29 Cost -5 Cost + 12 Cost is the required solution Note :- 6 The The solution of System X'4= AX (it) where A has eigen values ? 1, 22, 23, --, in and corresponding eigenvetors w, , Then solution of System of equation is + X ft) = Gu, en Note: O io Cos o tesind = Coso - esind Thank aking you