Question

Solve the BVP for the wave equation (∂^2u)/(∂t^2)(x,t)=(∂^2u)/(∂x^2)(x,t), 0<x<5pi u(0,t)=0, u(5π,t)=0, t>0, u(x,0)=sin(2x), ut(x,0)=4sin(5x), 0<x<5pi.

u(x,t)=

Answer 1

au 224 2x2 , 0 CX 51 &t² ocxLSIT ulot) -0 u (5( Tt) -0, tyo ut (x,0) = 4 sin su u(,0) = sinau, ut, u(x, t) = ® X(9) T(U) azu aru st² . = XT" =\"T Du? so XT" =x"T TU =a (let) T so T" x = AT - АХ Care-1 =0 I' = 0 T = at tb = catd 2 U = cat +6) (Cu+d) U(0,0) = 0 d (at tbl=0 - d = 0 EU (57,4) =) 51 clat +b) 0 =) C 0 - U = 0, Carrell 20 ut, a=m2 + B emt ..T" =m²T x = m²x mt ƏT - Ae = x =com mx + Dema Dow, C+D = 0 2) CD = 6 UD, t) = 0 U (Smit) = 0 Siim مر 2 srm ce + De u- o Case-111 aco let, a=-m² T" m²t =-m²x 7 T = A los mt +B sin mt x = 6 Cosmx + D sin ma.

i ukit) = (A los mt &B'sin mt) CC Cosmn + D sin mx) tow U(0,t) = 0 O=0. U(511, +) = 0 > D sin som o. if o=0 Then again 0 = 0 so, we take, Do sin sum=o= sin ni = nm ... 5 ПМ com = n/s. Therefore, general solution is text Σ L na O U (2, t) 5 B'D=1 D sin nx (A los nt & B sin ny) = (A los nt +B sin nt) sin ne (AD = A Therefore, general solution is Ž [ An los nt + Bn sin ht] sin ne U (2,t) = n=0 bow, U (2,0) = sin an = sin an 2 An sin nu 5 n=0 un + 10, Alo = 1 & An - o pow Ut (x, t) = in nx. + Br los nt nt I r [- An sin ne n=0 pow Ut(x,0) =B4 sin 52. 8h ท = 4sinsa. slu [Bn sin na na nco 25 B2B .. B25 = 4 = 20 25 4 S 5 요 Bn = o v n725 Therefore, U(K,) los at sin ak to sin st sin sx.