Question

differential equations

2. (a) Verify that yı = e cos x and y2 = etsin x are solutions of -2y + 2yło on (-00,00). 204 Chapter 5 Linear Second Order Equations (b) Verify that ifc, and are arbitrary constants then y = cre* cos x + cze sinx is a solution of (A) on (-00,00) (c) Solve the initial value problem y" - 2y + 2y = 0, y(0) = 3. y'(O) = -2

Answer 1

Given differerdial equation y"-24 +2y=022 d212 can be written as dy - 2.04 /f29200 Let yo nema ke the trial solution of equation 0 then Auxilliary equation (m²2m+2) =0 a m -2m+1) = -1 (m-1)=-17(m-1) = Will be >> m = 1 i So. General solution is ya) elfe cost of a sinx? where ei, ea are constant Now Y, (x) = e." cosx, Y₂ (a) = e sinx W(Yr Y2) = / eccosa ensina le cosuresina e sinnter eos = eixo.cosnté cosa-é?cosa.sma te?Six = 70 So, Ye (2), Y (2) are two independent solution . of equation o Also, y"-29, +24, = (e cosa - eʼsina resima -e "cósz) -2 (écosa-erima) +2 écosx 2-ésima - zemlosa € 2026mm +2 ecossai Similarly Ye also satisfy the equation ④ So, Y, car, and Y₂ (2) are Solution of differential

③ we then are independent solu solution ependent on of equation ④ ? We know that if too y(2), 2(2). Toho & Yo (2) + C2 g (2 also be the solo of the differential equation. There e cosx and ea sina are two independent solution of equatio solution So (ciencosa t easinx) also a of differential equation ... Y622 = ciécosa rezésima y (9) = 4 fé cosa - e" sina) +czlé sina te cosa) y"Casa elécosa resing - einareclosa - tez(esignare cosn te cosa -én) = - 2er Sinx + 2r?cosa Now y" (m) = 2y+zy (izgésimaet202 e cosa) -2 (elementary – 22 (e sinna en cosa) +2lcie cosa tezé? sina) s e sima (-24/+266-763 +70,) te?cosa (262-29 - 12+26) =é fina.0 + ecosno =0, so @ is a solution of equation o 1 h

② ② stion ☺ Given y_24'+2y = 0 -- 0 400)=3, y(0)=-2 we get the solution Y(a) = Geleosa of Cze? sina - y (29 = Q( é cosa-e6ima) tezlé fina té cosa) using Yeo) = 3 in equation ② : 3= er i and using y'(o)=-2 in eacuation ® -2= 4(1-0) + @z (0+1) -2 = a + C2 C2 = -2-3 2-5 - So the solution of the esuation is Yex)= 3 e? cosa-se? sina yeag = e( 3Cosa - 5 $'nx) Answere)