Question

This is a differential equations course, please answer all the questions.

Find the general solution.

1 - 1 8. I'(t) = 3 3 2 2 1 -1 (t) 3 2 -2 9. z'(t) -2)=(1) 10. 2'(7) = (1 -1)=(0) 11. z'(t) 2(t) =) =(0) 13. zº(t) = 6 (5 -3) =(0) 14. 2°(8) = (1, -1)=(0) 12. z'(t) = ON DINE 1 0 0 15. x'(t) = 21 2 1 -2 (t) 3 2 -1 16. :'(t) = -3 1 -2 0 2 -1 -1 0 (t) -1 0

Answer 1

4. ' (t) x (1) 2 2 1 4 ARCL) Where A = 3 2. 2 1 I- -1 4. IA - 5) 3 2.) al 2 T =0 (TX) raft (1-1) [41-4) (2-2) + 1] + [41-1)3 + 2] +4(3-2(2-1)] 5) (1-1) (-241–24+2 +13+ -3-39 +2) + 4[3-4+2)] = 0 = (1-1) [ 12-24) + (-32-13 + 4[4+24) = 0 > (1-2) [x² -7 -1] – 371 4+ 8A (1-1) [11-1] - 5+51 = 0 5)-(1-1-1](2-49 +51-5 = 0 =) - (x2 -7 -1) (+-1) +5 (4-1) I+ (1-1) [-(x2-x-1)+s

-(A²-7-1) + 5 72-a1-5 =0 -) 1?-1-6 = 0 > 32-34 of 21-6 y yo t (4-3) 42 (0-3) = 0 - x = 3,12 a = 3 1 2 =1 3 = -2 are Valuel TT Eigan of A

for X 3. x (A -x) x 1-d 4 2-) k3 2 4 4 T 3. 2-3 2 23 2 1 1 -1-3 4 4 3 22 -1 1 13 2 1 - 4 R > 2 R3+3R2 R3 → R3+ R2 -14 0 -5 x2 :) 23 O 0 - 5x2 + lox3 = 542 along = 223 X2 - 21 - X2 + 422 0 - 2.34 2X3 +443 - - 214+2x3= -) 24 2N3 24 X2 223 = x3 23 23

let x3 =1 V 2 for Xal (A-2)* :o 1-) 4 3 2- 12 23 o 2 1 -1-) T 94 3 2 - - K2 co 2 23 1 -1 4 3 000 22 no 2 . 2 .2 3 1 U2 O 0 4 23 R2 - 2R₂-3R2 2 -2 4 4. 6:0 (36) (8) R3 R3-R2

- 12t 423 0 커 기2 403 2죄 + K2 - 2X3 이 - 2의+ 423 ~2X y 24 4243 = 3 23 1 20 43 23 4 23 23 ht 3. 디 V 4 for A2 (A -Ali) 4 ㅠ 3 2 어2 2여 ) 3. 2 . 니 시 142 ) 4 ㅠ 3 4 1 M3 2 1 3 4 지L u3 (3) 2.

R2 R2 R R3 - 3R2 - 281 3 4. 5 -5 ( 23 -5 3 4 M M 0 5 -5 X2 43 5 H - 583 l2 =M3. 34 - x2 + 443 34 - k3 + 483 - 3er +343=0 = -23 19 - 23 22 43 23 43 23 uz 1 Vz 3) T V, V2 Ai, de, is respectively Eigen vector corresponding

Soln of 0 is nivou by. x (t) = arietet quiet & cauzet + C3 V3 e at + C2 Vz edzt & Ca va etzt X(t) G 3t e & C2 2 + at e + which is general soln of