Question

Solve the differential equation Try" + xy' + sy co, yCl) = 1, y = 2

Answer 1

y(1) = 1, y'C1) = 2 nay n +3ny'+ 5y = 0 Consider a=ez. Or, cognat. dy , dy .dz dh da ch my wory o) yndy - dy - du = dy ch2 d22 . Putting the value of ① and ② in the ODE de doy ) +3.( 44 ) +59-0 se +2 dy+3y0.- © So, auxiliary equation of 3 is. int + 2m +52 0 > m= – 2 – 54-49.1.5 - 2 -16 -- 2 thi 2 402) e é acos 22+ B SinE) = - +24 946) at (a cogfloga) of sinelogue)) Cen) y cetologe

Y() = et (AC 22 + B Sin 2E) y dye - - e ( A Gos22 4 B3in22) + f2A Sim22 +2 B 203 22) - A Coy 27 - B Sim22-2A Sin22 +2B ( 22 ) - A Cog 22 -B sinzz -2A Sinet .ot 2B los 22) 10 R Loga) - BSink loga) - 2A Sin (2 losse) 42 Tot 2 Blog (2 logr)? y (1)=1 & so, from 9 1 = 4 (A Coas (2 log 1 ) + B Sin (210g 1) 1 = (A. Caso + B Sino) y (1) - 2 so from G 2 = 1 (- A Cos ( 2 Log 1) - B Sim (2log 1) - 245 in (21081) +2B Cos (210g 1) = 2 = 4 (-B Sino + 2B Cojo) [2A07 32 = 2B . Azbal y (0) = s y(x) = (A Cos (20g x) + B Sim (2logne) √ ( Cos (2logx) + Sin (2 logre)] I