Question

Q2. X = ci cos t + C2 Sint is a two parameter family of solutions of the second order DE x” + x = 0 . Find a solution of the second order IVP consisting of this differential equation and the given initial conditions X (0) = -1 ,x' (0) = 8

Answer 1

Reccoute the equation with X=0 x"= de est) = a (rest) = (xert) rest sint - Cost to of Sorlutions find a is a two parameter family of the second order ODE X"+x=0 solution of the Second oodoo l up Consisting differential equation and the given Conditions of this initial 260) = -, *'(0) =8 Ang Given linear second adlar with Constant ODE S xHx=0 This is n the form ol differential equation Coefficients. ay"+by'+cy =o. equation ay "+by 4 cy=0 of the from ert. homogeneous selection a assume for an at ¿ax = 0 test =) 82 eat. est (2²41) =0 12+1=0 =) r=- the de ce cocos (t) +C2 Sint) Hi-1 = 8 - realhe a d = £i. for two complex yoots , to z tohemp 6, = +iB , ^, = 2-B general solution takes the from a seat C, Cos (pt) + Ez Sn (5:)) : 0 =1) El Caicost +cz Sint) 2 = 8 value Sub in zi gl - cost +8 Sint C, cost +Ca sint, Apply initial conditiona 200)=-1, x'60) = 8 X-cost48sint x'(t)--C, Sintt cz cost X(0) = C, Cos (0) +c2 Sim (Obja GC) +0 =) G =-) put x' (t) = -1(Sint) + C2 Cost -- Sint +CzCos -- Sin Col + Cz Cos (0) -) 8 = 0+02=&=8 X and s! (0) E8 value Sub in a gl =-- cost +8 simt Focastuesint