Question

5. (10 points) Find the general solution to the DE using the method of Undetermined Coefficients: y" + 2y' + 5y = 3 sin 2x.

Answer 1

15 Son y"day't arya 3 sinan 3sinan Given (0²4 ad Aoxy = is ma a am a 5 aunillasy This e quartier OC - 17 11 2 - -21-16 2 - 21 ci 2 -1721 complementary tinction CRECICOSoxteysinga? C and Co uncoc constants. Carbintain hrey special sinox ob conocspending Anal Solution , R.Hos ok &an loco Particular integral Tytucionit singa let dhe of vial coin rartitetom indego al be y* Since y ? ( -; ܙ܂12 ܝ rpeetlen I, we have . Day there of sy =) 0² (Alcosaxt Azcinang ÖDEMISO52 X AA z sinon) to 5 (Ariosar a da sinan) = 3 sinax NOU DY sinang diye chiles?*+M? - 2A, Sinan of 242 cos2x D²y* DIDY*) (-2Aisinang + Arcos29) -LAICOS 24 -4A2 sinaa w

COS2xt equation NOO we have from 56 -LACOS 2X-L1 An sin 2x + 2(-2A Sin 27 7 2421052) +5(Aicos2x + An sin 271) F 3 Sin 2 x =) - LAIlosan - 4 Aasinan - Lag sinan a 4 Azcosan +5AICOSax & 5Az Sinax 3 Sinox AICOS2x + 4ACOS 2x of Azsin 2x LAI SI'n 2x = 3 Sin2x sinex - IAIT 4A2) Cosan f (A2 4 AISA 3.fok = 3 sin 2x New equating coebbelce'ents ob like terms, we get Altlag Az - YAL 3 and new solving €99 @ LA1 &ltA= - LAI & A 2 12 AS 3 =) Az = 2 h niele so 1 m 2 -4* ald SCO I? cosan + :3 Sin 2x 1 8 - Frem Equation 2 and (Sj we ) set the reequired Solution of equation رش 1 ч (1) (C, cosax of Casinan) 口? 3 sinak 17 ja) cos2x lege-x + 3) Sinax (ans)