Question

solve the fallowing system of equations step bu step

tos) Resolver el siguiente sistema de ecuaciones X' = G )x+(sec ): para x(0) = 0 ,y(0) = 0

Answer 1

let = (%) x's ()- 60 :) (? )+ (Sect) Centro (19#sect) x = -4+ Secct) and y'=x. Cor) ds = -Dy+ sec 4) Dr = -Dy + Seccth:tan(t) and x=Dy ie -(1) XD x = DY CD716 = Seccto tant) -(3) The auxiliary equation of 3) is fmizo ie, m²+1=0 m=zi. . Xc= C, COSCt) + sin(t) u=cosct), =sin(t) and R=secit) fanct) u's-Sin Ct). vecos (t). let cos(+) Stofte = coạetn+sip?zt) u -Sinct) =1&o. f(t) = -fre de sinct). Secttitan (t) dt

f(t) = - Stant) ot Sci-Sect)dt - t- tanct) get) = furdt - S cosct) sectt) tanlast = Stanctdt = Insect) I Rp= uct) f(t) + v Ct)g (t) = cos Ct) (t-tan (t))+ sint) In seceto >tcosct) - Sin Ct) + Sinct) In/Seccto). Thus, x = Ret Rp H = G COS(+7+9 Sin (+)+ tcos (+)-sin(49+Si76) In sect! and y = Sect)-x' = Secct fasinet) +6C0s (4) + Cośct)—tsm (4) -COS CACOS (4) bo sect) tsinkt belttant secco

sexe tant y-Secct)-ze! = Sect)- |-4s1064) + GCOS (+)+1.cos(+)+sin(+66S (4) + cos Lt) lo/sect) + sin(t) Isecttat I seething = seclt)+csin (+)- SCOS (t)+Eosinct)-cosct) In sect] - Sin C+cion 260)=0 ie, C, C1+960 10-0+0=0 le G=0 96=0 ie I+G(0)-60(1870-mlol-0=0 HO-G-0=0 [caz Sinit)+ tcos(t)-SinCt) + Sinct) hos I seet) X16) = tcos (+)+ sin(t) In Isecct)). y(t) = Seclt) - COS(t)+ +sint)-COSCto Insect)- sedtitan (t) alt= Hence, x= tcos (+)+sin(t)ln/secct)) Sec(t)-COSLt)+ tsin(t)-COSC) o Iseet1-sinttant