Question

24) Solve the ordinary differential equation: 1 + 2 dt? dt + 5y = e sint, given that when t=0: y=0, and x = 1

solve
please as soon as possible

Answer 1

Ariswer! Given dy etsint dur +59 dt? it is non homogeneous OPE since y = % + Up V particular solution n To act un by dy + 2 dy +5y = 0 ता is given dra corresponding auxilary equation m2+ 2m + 5 =0 since mi-172 routs are imaginary Una part ca cosBt + BsInBt) :. Uns asrl, B f CA Cosat + B sinat) 4

I'S method of undetermined To find Up: Geven ODE of the form + qeny : 4022 dt ert sint Hical) ule can find up by dy dy + perd? dta cu-efficient from reale etsint ODC in Ypa et ce cost + Dsint) substitute Че the quen 4p"+24p'+ 54p toint flccost tosint) + teatresint +000st) e .. ye' 49": é cctostapsinh) - é (-csint +Deest) et (resintt Dcost) tent (accost-Psont ) 4" -22+ C-csinta deost) -soc- csont + p cost ) +91-8&ccost a osint) + A C-csend #pcost) ] + corticcost + ipsint etsint एर sint (+20 20 -20 ] + cost (-20 – 26 +21] = er sind By comparing co-efficien le of? sind Ę cash 20. G -20 l, Dry ер О

tsint Yp --- y : 4+ up etsint (ศ yi étracusot + B51006) - To find A & B lylo Given 4109= 0, de dy.-efracosat + B sı'nat) +e+(-RASINDA of 90 COS2+) thetsint etcost af 410) = (A -(0) А со 1 을 2 dy (0) = -1 (0+01+i (20+213) ar 213-5 - 1 B 22 / .. -t sinta of - 30 sinat sinat - ابا sint) yo et