Question

(b) Consider the matrix differential equation for the vector x(t) d dt - B2+ where B= (69) 4 10 5 -1 (i) Find a particular solution to the matrix differential equation. (ii) Evaluate exp(Bt). (iii) Find the general solution to the matrix differential equation. Express the general solution in terms of the components of the vector (0).

Answer 1

(2) d x = Bat Where U- Soln General solution- I 21 [] where s 242 H'a 22 [ TILSE}] X = 401 +102 + 541 - 12 + 4 let x = DHI and H2=0Hz DR = 44, + 10 12+2 (D-4) 2 + 10 +2 = - ( D2423 52, na ty 541 +CD+ 1)2 +4=0 - 123 multiply (21 by (0-4) and Adding in equation (1) 5 multiply by ean.es 51D-4)21, +5x1042 +10=0 and 5(0-4) Hy + (0-2 (D+1) 72 +4 (0-440 → [5o - (D-4)(D+1) 725 + 10 -6 (D- 40 -440-49]itgt tro-*#*#46** [sor (D²+ 10-40-41] H2 + Lot 16 50 [5o- (D+D-4D-417m 26

[-D2 D4 4D + 4 +50] 2=-26 [-1² + 3D + 54] 3 = -86 [0²-30-54] 42 = 86 = { g²30 454 [D-3D +6D-54] x2 = 26 [DCD-9) + 6(1-91] = -26 → [ (1-9)(+6)7 42 =:36 [laze, e 9t+ gerbt + 43 +26 p. z Geot 26 10-1) (0+6) (D) (D+ put the value 112 by en. (2) SH+ (D+1) Gegt + Cebt - 1 3 - 4 =0 50+ 9 czegt - 6 Cae-st + c elt , cebt + + + 4 =0 512= - 9e, e et t-gelt +6Celt - hebt u + 13 54,2 – 10 c est + 5 Cze-bt [1922 - 2 Gent + ca ebt - 2 - 2x] 2 209 27 ci particular solution to the matrix dift. AN H= 209 95 24 * = 7 7, 21 27

(1) 0-1 SJ This is part of G much larger ODE problem and & went to know how to compute Cup (Bt). calculating eigen vectors of B corresponding to the imaginary eisen valles. Then construct the matine P consisting or the eigen vectors as Columnis.

9 Cup (t)= Pemu (Dtpt Where the diagonals are the Dis diagonal matrix eigen values o 0. then eigen value of B 4 1 0 thou citon valle (4-23- 12 5 أم - 1 then B= {4-d) (-1-1) -5o] then eigen value (de 9 and dan6] eigen vector d=9 put the B then 4-9 [59 ] [-]-[o -1-9 -5m + 10 42=0 501 - 10 12 20 21221 5012 to y = 2 let [m=; 21,1 put the then uto [ -L+6 16M +1012 - SH3 + Gh

mutt 11102 and ADA (A) 10 Hit M₂=0 LO 212 +8H=0 Ans. »-a ند a cup(Bt) [ng=0 to groft S , S goo&roll, Štra 812) -40)dna 1 [ [ ] 8 Jana 7 ] CE el. Ilg II Beaf (Bt) 2 T- eup (it) = 7 - - T- id nay a Pub Where Id(70) arad 217 Ell drea [T. - cik pub T 2526) wayt - THE orut TK

solu to the matrix dill. cau. (iii, the general [= c est tu e-ft + 13 ] and [1122 - 20, e it + cz e-6t 109 the components of the vector ulog 970) AT – 2 C e + e-610) - 209 27 glol naz с, е ta &- 660) + [co-e-0 =23 then { -24 + C2-109/27 [222 + y + 18g] CD