Question

1. Find a solution of ay ay 022 + c at2 = 0, where c is a constant, that satisfies the conditions: (a (b) y=0 y + 0 y= a sin(2cm) + 2a sin(cr) when r = 0 when t +00 when t = 0.

Just do part (a)

Answer 1

- T do 2 072 I we have given. 2 c 22 0 0 To can be written as. T Tout c² Ytto (2) here y is function of a st. . let Solyton ot equation (2) is. y Co, t) : X (04). I(t). - Yan = X". yuxuI! Put you & ytt id equation (2) " from equation (2) we get. X!!, I t ?X. I" = 0._ =) X T = -(²x. " Ixil وقیمت آست 5) and X! -c?dx20 - 2) I' d I 50 =) I!+d1=0. -(4 case () when d:0. Then x" - c²(o)x=0. 3) xllco on in te grelting we get X = An + B . & ad T = 0 = "=o. an integrelting we get I c t + D .

case (ii), itd0. the has x c² dr=o. solution.x=A, e जगतम. + A26 8. I" + d Ico. has Solution. I= c, cosat + C₂ sinat = y(at) = (Ai perantare cray) (c, cosrattasin vat) case (lii in dco then x' c dx = 0. 5) x''+c?dx=0. has solution X= Ei coscron + E₂sinctan & I" +TEO.ST"-d Tso. . Vt. has solution T = Fight + fed = Y (H, t)=CE, coscant E2 sincrdu) (Fieldt fre

yco,t)=0 how we have given. y so when x=0 =) y cot) = 0. by using condition in XCH). T(U). = y cast) y cost) = Xco), I (4) = 0 : X(9) .(1). 3) X (0) = 0. from case d= o. S 8(o)=0 =) XCO) = A (0) +B. = O -B a B = 0. X(X) - AH where A is arbitary. = xow An .YCH, Z) =AH. CCE+D).

. from case d o & X (0) = 0. X(0) = A + Az : - 3 0 =Å PA2 5 A5-A2 Foto. 7 X(a) = . - Az e taze * XG) = A₂ (elevat . 1 (A, 4): A. (** *- Lê (c cos It + C, SII) vật)

from case don & X(0)=0. =). E cosc daco) + Ezsin ch (o)=0 = E o • Xcon = Ezsincran where Er is arbitary 1. y(at) = Ez sincran (fent + F2 et which is required solution