Question

(Find the general solution of the following systems of ODES n(32 0 1 3. y (t) A y(t), Aj 0/ 4 4. y (t)A y(t)+g(t) (0-1. qlt) Aij 2cost - 8sint/ 4 Please show all steps with explanations. Thank you

Answer 1

3) Consider the トマート The chara cteristic equation of A is JA-I0 - eigen valuas of A coe The Consider tho systann (A- )O cohen -2i ce have (A2i I)o 21 2 From ISt YOw. 2i8,ーズ2=1 eigen vector メi cohen>iニ-2i 8 The for a 21a 2 Cohen 入=21the eigen vectoY X2= -21 valua -2i and eigen Fhe Soluton to the the covres ponding eigen vector x s -2it Sin (2t) C(2t) 2 C (2t) 2SIM(2t)

The general scu-tion to tra syslam y'sAy -Sincat) L 2 cs (2t) +G 2 5in02t) (4) Consider tha suystam AY 9t) FSin2t) +C2 The Complementany sfution s 7, ct)- G 2 G8(2t) 2Sin(2t) The feundamental mabix s Ces(2) S2t -Sn(2t) tren L-Sinet) Cs (2+) 2 Sn(2t) 2 C8(2t) Compute the product 2 Cst csc2t)Snc2)T dce).ge) 1 L-Sinc2t)C(2t) 2 C8(2t). CSCE)-4Sin(2t).SiつCL)フ 2 cCE).Sn(2t) 4 cSC2t). Snct) 2 C1-2 S0t) caCt) 4 (2Snt cat)- Sint 2 Cest C2sLtst) 4 (2 cat -1) Sint 4S19t cst - 8 Sint Cost 2 C8ヒ - し-4CeB'tSiっt-8CBと Sint +4Sint 2 cBt -12 Siっt Cst 4 Sint 12 Ct Snt Compute the integral 2 Sint- 4Sint -2Cat-12Sin'ヒ csヒ Sce-gct)dt dt = 4Sinヒー12Cat Sint -4 cest +4 C3t

The ponticedo1 sdution s FPCEPRPC-gc Ct) -Sの2ヒ1「2 のつヒー4Sint 2 SinnT 2 C82t -4c8t +4 c83t 2 C8(2t). Sinヒー4C82tSinヒt4 Sin2ヒCst14 S02ECSも .4Siつ2ヒ. Sint-8 Sin2tSinヒ-8c82t C&ヒ+8 Ces2t8t The Sofu-hon to the -homogeneous equation y' AY+9) non nt-2st]4 L4Sna)[Sint-2st]-8cacaost- Ct +4 Sin "2C8(2t -Sinc2t) C2 2 C802t CC2t CV 2Smc2t) +