Question

Use the substitution t = −x to solve the given initial-value problem on the interval (−∞, 0).

$4x^{2}y"+y=0$ , y(−1) = 6, y'(−1) = 3

Answer 1

The given equation is, 4x²7" fyco, y(+1=6, 7(1)=3. 71). given that t=-x dt da 17 -1 nart dt dre dy ar dy dt = dt d dx d q dt der du YN so *=1 at x= -1) eer (1)= 6 Tas tol at yllu = 3 Hodno tark, the equation (1) substitution hiso the becomes 4 t ² d 4 (6) = 0 7 (4)-6, 7 (-1) =3 d13 (2) This is and order Homogeneous equation. het Z = legt ( ૨ I da तर / dar +2 dy da log T Now, de 22 de dt AP E d dy. 6-142) + $ di der at - de la oy oy where 29 dzz - 016-17 o=d2

general solution of the equation (3) is, so the equation (2) be comes , $ 4010-1) + 1] -0. 0, ( 409-40411720. (3) Auxilary equation of the equation (3) is, [J= A e me / 170] Aman Amtl=0 m = tik so the 옷즈 12 y = Ge + C₂ Z e A legt logot to lost e log it bo VF qe of C logt e quest blogft =GUA + lg. log tot arala fC2 loge). Grela log 41202 ylt) = (a rele lood). I to of st. art Now and H11)=6 implies, 6 9+(210 =41-6. y (1)=3 implies, 3 (+212 =) 262 = 6-6 =0. 2 so the required Solution of the equation (2) (= is, HA) = GUA where -00 LALO (Answer).