Question

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

Find the general solution.

9. z (t) ) = ( ) (1) 10. (t) - (TEA) (1) 11. z'(t) = (t) 12. 7(t) = 13. z(t) = ) (1) 2) (1) 14. 7(t) = (, 2) (1) 2.) AC3- 10 0 15. z (t) = | 2 1 -2 () 32 -1 100 (t)

Answer 1

X'(*) = (1 -3)*67 9.9. Solution : we Assceming solutions of the form Quel ecclue the asysteni rout (] [E] -[:) ] 0 for cégen values a and eigen vectors Z for a the materis ineuttbe csengeelar. 3-8 - 2 PIC = 0 4 -1-8 -348-38 +224820 = (3-3)(-1-7)+8=0 => g?-27 +550 3) ) 7- at 54-20 11 2+41 11 1120 2 when r= 1+2i 3-(1+27) -2 -1-(2420) 9-2 -2 O 4 -2-2 O R, R, * ati 2(2-1) (240) -2 (171) -2-2 1:] 4 P) -2-2i 4 | :] -2-21 8 ] = 42, +(2+2) 320 ziz [ -i] [si] We only need cone elŝen values and arzen vector pair, if we need Zz othen 2 = 3 Note that

1-L 요 L-i I cos2t - shirt Suice %2 = 89 , then one solution is H, 4) etziget et (cos 2t + Esnat) et suit co2+ Cosatt smizt Thes, oue general colectiow ced ne (*) = G, et Cosat suit ezlet) [ Coszt+ suizt Coost-suizt). Qdo. *'(+) = (11 -) alt) + + ie [ csolution 4 I -1-7 for 12:1-1 for eigen values & and eigen vects • the nation must be wingular 1973. Al- +(1+)-430 (3+1)= 4 - > ra tat2 a 80-3,th when +1 is L-(+) 4 I falte) 요 4 ~2 ㅗ 1:] [1218] Ri -k Ri

Z, -272 = 0 z = [?] Cohen &=-3 1.-(-3) 0 [1.600 0 -2-(-3) 3-1 [ 1 21:] of S 2, +222 مل Qen Zz = [3] Thus, the general solution is 2CH= C. [3] e* +612] @ 3+ x'(d = ( xe [*] (3) [:] -0 for cigen values & and eigen weaters 2 for the materib. must be sengulas Solutions - Qax 5 - (2-2) (2+2) +5=0 place -) -(4-22) +520 -4+22+520 ♡ s. g2+1=0 =) &=ti which y =

[ 2-i as I -z-i 10 ] sorto 2-i as Q- -(2+0)(22) ע - 1:] the ai -( 81) ari -5 c [ 2 - 1 2-1 -5 18] (2-1) Z, -52₂=0 1 Hence, z = [ 2-1] as, 2-1 Note that we only need our eegen value and eigen vector pair', if we need Zz other 국 : Suice = 5, the one solution is Z2= His کی Jecit an, [e i Je [ee] eit [at] (cost + csuit) ricost 2 Cost -suit t لا sunt 2 cost - sunt sunt Thus I our general solutiai us

Hld) = Cos cost [ 2 cest - suit et G D sunt 2 cost tsuit (*** Q. 19. zy) 2 -5/2 alatt 215 solution :- Suice 28-562 ] [2] [3] for for eigen value & and eigen vector 2 the matrix must be singular ३-४ -S 215 -1- - (278)(1+3)+1 = -(2+28-7 -72)+1=0 > 8+82-9+1=0 o=1-erre =) at- ettega Orr-ere e 87 DoTS ET 13 1+3 2 12 hoyon - 0 8= 2 +536 2-(4+2) 5 :-( 14UP! O she LE!) 2 3-536 slo al | : ] -3-16 R₂B I (20830)

3.531 -5/2 E samante - (3+52) (2-0 1 : ~5/2 2 Pa 3-58 3-530 o -(0:2) مس 3-S3i 2 2-13 -sh 1 : 3-331 - 2 C (31)+ (-5)2 = [as .. then 2,= Now, note that we only need one eigen value and eigen vector pair if we need ?q=ą z ittien chuice rozši then one solution in Mia 1 e (4+52) it 3-9 [34 eb ( cos s3kst + it tisui 52/ 3-530 t 3 [ = eht Cos 3/4 t 3. ross Sat - 83 suis * sur 1₂7 3.com 52 2 + 3 sinify, at leht [

Hence, our general solution is alata C, et Cos5²7 3 cos by a B suit & C₂ etat fm?fret 3 cos syut th suiszt be solution?
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