PH12130 Problem Sheet 2 1. 9. du-2den = 0 at Jx2 let (xit) = X(x) T(+) (0,+) = = = > x(0)T(+) = 0 :- x(0) = 0 ((Lit) = = = > x(L)[(+)=0 .. x(L)= 0 24 = XT' 2= 4 = x"T : XT' = "X"T then separate variables T' = x" =+ 1 XT X => T' = +x XT T'(+) +? XT(+) =0 =r T'(+)=?AT(A)? => x" = +1 × x"(x)= >x(x) = 0 ~ x"(x) = >x(x) b. T = A+++ c. if P=0; AT=Ax+0 = A.1 = A d T= Ax+1 ; 130 .. as + increases, T increases forum som sonomaterial soty X = Broskx + Csinkx where 2 =- k2 which is unphysical à ignore scribble Anowant page for parklar 1. T=Aex+1 ; ><0 => > =- k2 .. T =A .- x++2 ; T = ACKTHE n = 1, 2 , 3 ... .. no variable the is non-zero .. non-trivial solution.
g . XT' =xx"T = > : x"+ XX = = T' + XxT = 0 > = - K2 ; k = nr L T = A= x (2)+ X = cos (WTLX) Using principle of superposition u(x+) = Arcos porEx)é 00 (vit) = [A cos(rozx)= " (+)} na, w(x+)= a+[ An cos(mcx)=(47)} 4. An = ? (e) = { A cos(2 ) 1 ..... = 1 n=1 1 telcos/ms dr = [ B cos(nick cos/mIA) dx = = Sum ={l i n =m 0 A=1 infm C = Z Jo An Cos (IL) CO (MA) 5 = A. (= + 0 +0 ... ) 11 ? N .. An = 2 [ sim (=) flal de
i. u (x , 0 ) ; +1x)=c/1-cas/2tx L 1) An = 2 1' (1 - cos mA) cosunA de = 2 [' (1-cos e X) cos nus dlx = 2C / cos Rx dx - 2c / cos 2Mx cosnix dx E !? L L 1 ? L 0 = 2CHTE SinNTE -