Question

Question 1: (20 points) Find the solution of the initial value problem a = cos? x – sin x – 2y cos x + y2 , y(0) = given that yi(2) = cos x is a solution of the differential equation.

Answer 1

die solution . To find the solution of the given Ydy = Code - Sime - 29 68 x + y? 760) = ? given that Y, (a) = Cosse is a solution of the differential Equation, So, you = Y, (a). V(?) % = . Ulaj Corse is the Complete bolestion of differential Equation. so, y'la) = Vlla) Goset U(2) x (-shine) So; y'1a) = null) Cafe - Ula) bina. so, ie (2) CoJp = v(a) line = Costa - sina :-2 (VIX) Gya) 637 + Tuia) Cy?) So, blan) Gjy = corall + vita) – 2 Vla) † (via)-1) time) . So, let o we have fumes 1.609 ve = Cap Se (1979-) + (012)-1) hisze Cel : 0(X) - | - Yo (2) So, y (7) = Cose y + y sinne

d2 dyla) - 9. (n) tome = 9:12) Gyep. son las con est une = 654 Cep i la = 4(2) (Yalaj)? da da So, dofta) + f(a) fone = 637 this a linear differential Equation in 187 oscles. So, I.F. (In legeating face toa) = p I tone dopp (n [tca) da - 827 So, Al2). 8ec (P) = 8 Grp. $ec# de te o where c = Constant So, #la) bec? = lide + 7 1419) = (3P + c) GAP

anel, (2134-1) it = +12) = $17) (12 ) = Un)-1) So, 12)-1 = " A les vlt) = 1- tres So, jula) = - Photos Corte 904) = lasa li- per pce) carta 9a) = GIRL 3 900) = 7 9(0) = Co (0) - Topic so, 2-1-} 4 2 - < 1 105-2] So, lyraja Gora