Question

II) Find the general solution of the following ODE of the 2nd order using the method of undetermined coefficients (25 points) y"+2y' y 2xsinx

Answer 1

Coefficients Ans - Wring method of undetermined to find the general solution. Given ODE is Y" + 2y't y = 2x sinn Rer writing eq. as dy +2 dy ty= 2n Sink da2 + 2ax +y = 2n Sink - (02+ 2D +1) y = 2x sind, where Ded Auxiliary equation will be m²+2mfl=0 (m+ 1)2 = 0 m = -1,-1 Complementary solection Yc = (C, + (24) &* where Cig (q are any arbitrary constants How let particular solection be Yp = (A + Bh) Cosx + (C+Dx) sina so that yp = - (A + Bu) Sinx + B Cosh + (6 + Dw) Cost + D sina and yp=-(A+BH) Cash - B sinn -B sinn - (C+Dx) sina +D Cosse +D GSK

- Now Yo" = - (A + Br) Gosx - 2B5nx -(C+Dx) Sinx + 2 DGSX satisfying values of Yp, Jp Ip in eq. O, we get - (A + Bx) losk - 2 B sink - (C+Di) Sina + 2D Cohn - 2 (A + B) Sina + 2B Cosx + 2 (C+Di) Cosn + 2D Sinx + (Af Br) Case + (C+Dr) sinx = 2a sina - 2 B sina +20 Cosse - 2 (A + Bx) sina + 2 Bloss +2(C+Dx) Cosk + 2D sinx = 2 x sink (-2A-2 B + 2D) Sinx + (2B + 2 C + 2D) Casa Equating & Cass - 2 Br sina + 2 Da Coss = 2 x sina Co efficients of sine, Cash and a sink a we get - 2A 2B + 2D=0 2 B +20 +20=0 - - 2 B = 2 2D = 0 from eq. 6 , we have from eq. 6 we get [B=-1) [D=07

How pretting values of B, D in eq. , we get -2 +2 C +0=0 26=2 How putting values of B, D in cq. ③ , we get - 2A + 2 to co - 2A=2 - A=1 Now pretting values of A, B, C and in equation & we get Jp = (1-x) Coss +(1+0) sink Up = Cosn - scoss & sinx General solution will be y = y + Yp - y = (6 + (2x) ex & Cost - uCost & Sink