Question

1. Solve the nonhomogeneous linear system of differential equations d / 8 3 1 / -10 25 u + dt" ( 3 8 ) | 10 l -25

Answer 1

0 )u+ (-19)+(25)+ > first we find eigenvalues 1A-dIl=0 8-)(8-)-(3) (3)=0 64-82-84+1?q=0 x²16x +55=0 2411-54+55=0 XCA-11)-5(4-11)=6 (7-5) (+-11)=0 x = 5, 11 .:)=5, Ja=11 Now we find eigen vectors: >,=5, we'll need to solve (A-2), = 0 R2 R2-R, R, R1/3 61) => 21+=0 =) 2,2-27 put x2=1, then x=- 2017 (6) 12-11, well need to solve (A-227) va = 0 -3-319) RaPatri (-331)

256 RR, 1-3 6 :19 >) 2,-*=0 ) 2132 put =1, then 2011 --Va= acce 5(-)+czet (5) Let up: (61) ++ (8) up = (a) plug it in system 6) (33)[06:++$,]+(-)+(45)+ $ 3[+62)+ (50)+( qa, t+80+ 3b,t+3b2-10+25t 39,0 + 392+86,t+852 +10-25t sait +8a2+ 3b, t + 3 bz lot 25t-a, 3a,t + 3a2+ 8 bit + 862 +10-25t-b (t (89, + 36, +25) + (892+ 3 ba-10-a ) ) t(391+86,-25) + (399+8b2+10-b.)) (89, +36, +25=0 and (892 +3 bg-10-9,=0 391786, -25=0 (302+8ba+10-b,o 8a1+3b, =-25 - 8aa3b2-a, = 10 - 391+86,=25 392 +8 62-b, = -10 » arattaan 8L bit +bal - 25t

we have (891+36, = -25)3 (391+85,= 25)8 24 94+ ab, -75 244, +64 b, 200 –55b, = -275 bl=5 Sait 3b, = -25 :9,=-5, b,=5 89,+305) --25 sa, +15=-25 89,- -40 -) Q1 =-5 from ① and ② 892 +3 b2 -9,10 =) 892+3 ba -(-5)=10 3a2+8b2-6,5-10 392 +852-52-10 saat 3 b2+ 5 = 10 3a2+8b2-5=-10 (89&+352 = 5)3 (342+8b2 = -5,8 2482ab2 = 15 2402+G4b2 =-40 - - G -55b2=55 = ba=-1 892+362 = 5 842+36-1)=5 = 8a2-3-5 => 892=8 Up = ( t +(-1) gen.sol u=uct Up .: U=c, est (-1) + Gelt (1) + (-) ++ (-1)