Question

6. Use the method of variation of parameters to solve y" + y = sin(x) 0918

Answer 1

Solution :- Tue given dieforenlial equation is, y"+y=sins - ③ Now, We Selve me dilexetal equation by method of Variation of Preescometers. auxiliary equation of y" tyzo is, m²+1=0 where you is a solution of y'tyzo of met Hence the complementary funcion of is, Acosm + B stir where A, B ore Comstor , Here, cosa and sure are independent Solutions of the equation yhtymo as wronskian of cost and sein = I cosa sunt I-suin casa! = 1+0 Set is choose & and U2, both are function of a in such a way that © is Gatisfis completely by, yen, cosm + Uswisi Dukseverhiahng both sides w.xit y=-6, sunt u cosm & V, cosm & I sinn we choose & and a such that an Cosm. &,'t sin. I=0 w ur y = - 8 sinn + O cosm and y" - i casm – Vesium - Visum + Ulcosa Putting this value of y, y' and " is the given equation, we get, - ,'sci® + 0,"cosa = sma - All or - Olsun & Cosm. Di cosm z sina Sina os, -1. Simon a V, - Strefa and d = – cost. sun u. 86m) = f-sura de Sins (from@ ] =-(0-cosa) suunda = sinn cosa cosx-cost to ona v cm) = sua cosmola = Sim +

Hence the complete Solution of the equation is, Y= (cosa cash + a) casa + Coruna t om = 0, cosm +G soia + Sula-costa + casa = a cosm & siin – Cosa cose 37 where ea and ca are constants,