Question

8. Given the non-homogeneous linear system of differential equations x1' = -2x1 - 7x2 + 3t X2 = -X1 + 4x2 + e-6 a. Find its homogeneous solution using the eigenvalue-eigenvector approach (10pts) b. Use the variation-of-parameters method to find its particular solution (10pts)

Answer 1

Solutions se = "249 -742 + 3t Dear = -3 +4964 tengt rare 204 X -] 31 2 7 + :: x'it) 204 218 Lp X(t): [2] 2 -7 30 xt 1 66 compare with x!(6) = AX + g(t) 36 .. f get? of eigenvalue f eigen reeds @a To find consider 1 A-a 110 -2-8 1 4. (-2-x) (60+) - 720 g2.20-15 20 (443) (da 5720 .* id=r3,5 doe eigenvalue of a

Now, consider, [A-91]1: [0] Can JO17:1:1 for 3 [ 31131-6:7 Pag - (. :71231-(:7 .. 282 is free variable LEIR Pet 212=t 201246 Tare tid 0,- - [7] is eigenvestor of a foo =13 for das 6:30:2 1:16)-(:) Best Pet - [:])[:] A la ů bee variable al put Ha at t EIR 2-t

Dare th ... V:[ is eigenvecton o A for 2:5 .. solution of homogenous equation -35 X(t): ce 29(7]+6**[:] ] it xct): cent ".7 + (-1) Caett + Caest ele 36 7e16 -esh Date W est e.it est 1918 Feat teat seat WE 8eat est est .: Wie seat erit 7er3t est e36 'P e] est 7e-st 18 Wrigita 35 e 3418 L-e-st 18 eatl8 gestig (]2 er Wriga 3 test .18 té 3k 18 -Breast 7 elit + 8 8

sulg dt = [ (stest e f g de lit I ft 3 test 7e + I dt 36 1 [est (34-1) --] Surig at 3 est (St+1) - 7ellt 200 88 sp au J4 lg dt -est koerst [est (36-1) -] Kpitia 17 e 37 erst est 3 doo est (5t+1) - 76 alle 88 -64 7 e + Xpit la 24 200 Ž, 4-66 [erer 37-12-1]+3 (56+1) amere [est (16-17.–1] + 3 (544) 7 east 88 200 200 번 87 7e ole 88 soll of Xpiera [ 4[13t-)-8=64 1-? (st+1) + 7 prot de C63E-12 - erot] + ako.(5+1) .. The genesal X(2XcCt * Xplu Crest + 4[134-1 )aport] 2.. (t+1) + Lent to mo [134-1) -est] to 154 +1) Eine rey frie it X(t2 35 cie thest

се Xilt=7 Ge BE 200 88 st+4[634-1)-erot] -2 056+1)+78067 da [C3t-1 ) nenot] est+1-7 erk 3 7 Xalt2= Crest teleskt 24 200 88