Question

Find the general solution to the linear system of non-homogeneous differential equations x = x + x + 1 xz' = 3x1 - x2 +t

Answer 1

al=alltolat (1) Question the given dineal system of how- homogeneous differential equations is (1) and ol 30-5latt (2) Solution: Equations (1) and (@) can be weitten as Gi-Ola and -391+4+6= -di-da dala and -bait + =ť dt (0-1)0,- 4 = 1 43) and –30, + (D+1.9 = 5 -14), DE Oberating equaticom (3) by the obecator (D+1), we fave (0+1)(0-101,- (D+1) = (D+1) (1) (DP-1)0,-(p+1)% = (D+1)(1), ca-biçard) => 102-1)0,-(D+1) Q = { 20+138 of at

(२) INA Itt (02-10,-60+1). = Coti? (DP-1)0,- (D+1) Dla = 1 45) Now aidding equations (4) and (5), 'we have (D1-D4-341 = FID_1) -330, = t +1 (824) ali (E 40 16) Now equation (6) is a lineal differential equation with constant coefficients. .: The querillicly equation of the diffeleytial equation (6) is given by (48-4) (Wi-99) = 0 = (---) ( m+ ) = 0 (4-6)(4+6) = QP62) q?-6) W = -2,2 Loots of the quailliagy equation (7) ate teal and distinct, therefore the Complimently function (GF) of the differential equation (6) is given by ciest to eat 48) Where C, and G are arbitrely constants. +7) ( Coro

(3) Now the particular solution (1) of the differential equation (6) is given by foldp = jpi-4) (t+1) ܚܐ 4 = D4 + (t+1) (-4)(1- ,( Note) 411-0) "CE+1) #( 1 +(?) + (83)*4 (t+1) : (l-a)/ = 1+0+32+93 + by Binomial theorem 1+ + ---(4+) ${(t+1) + 48*(t+1) + 1 D4++) + (t+1) +1) + 10 +o) + foto)+ 0 to D"(t+1) o for n>2, leo (t +1) = 0 ath = -f(t+1+0+0+--- - (t+1) T6 4 91 16 up for no

(4) (t+1) 4. -at o's di Се 4 at e a particular solution of equation (6) is alle +9) Flom equations is) and (9), the geveral solution of equedion (b) is given by Cofo + fub +6 @nat (t+1) tlo) Differentiating equation llowith lespect to t, we have ali २८, हरे -२ (1+0) (11) 4 Now feory equation (I), We have ai-oi-l Ef-2c,ett +26 et 63-scienteret (E+) -1 if by equations Cloy and civi) +24 24+ [ -ciēst-Get Get (t+1) =-307ēxt + set ما -at -2ci eta -20ězt+260 4 -1 + € *

-| 4 we have -39, ērt + Gent + .:92=-30, ēxt +G -3c, 8 at & Geat (t-4) -TQ) .: Fromy equations (ID) and (12), U = ciēåt + Goat (t+1) and 04 =-348*+Get+ (54) is the general solution to the given linear system of non-homogeneous differential equations (l) and (2) 4