Question

4. Consider the nonhomogeneous linear system of differential equations / 4 3 4t / cos(3t) + 2te4t / l - sin(3t) / + 4tºe4t / sin(3t)) 43.4t sin(3) ( cos(3t) ) Given a particular solution t²4t / t th 4t / Find the general solution of the nonhomogeneous system. Hint: det(A – XI) = 12 – 81 + 25.

Answer 1

Led AT (-34 . 2 T- 1 given Characteristic eq" ir det (Art) -%81+2500 :: 87.1 64-100 . 2.&3614£3i Let X (a) be the eigen rector . corren ponding to dizuti. Then - Ax= 0, x giren . (4-4-3i ).m + 3h :-o.. A -324 tu-4-35)ų to. the equivalent system in -3i vi.+ 3x2=0,

or imam on 4 - m - r - e rcinomay e I non-zero complea on ere, kaic no.) choostra e Mrl, mzi Then x"=(!) Similary an eigenrector corro exponding to the eigen value dza uo3i in.. xYclu. General nontion of correspon- ding homogeneous ean in (4+31)6 ,, (4-306 и, (0) - че s40" (esit) tezett ciprian . ut costtijinat ut illorstaisin3t)) - aue Cicloszt tisinges the coszt- isingt) a 4t / cartrigint luht/ sinatticost Aulna ce l-sin3t tilost) the ( cost-isinat • T ne ut osit I

. on heritem S e - on General 2014 of nour homogeneou hyntem is ulti= Unlt tuolt) ult . 4t/costrigint le ub/sinsticoszt Sinatticos?) to (cost-esint 4,4t/ Cor3 1, 46 sin3t I Cosz a reaf t: Upit are giveno G e - (-sin3t tt