Question

3- Solve the ondulatory problem on (<2<1,t> 0 un +1:/3 = Ur. u(2.0) = 1. 2: (2.0) = 0. u,(0,t) = 0, 4,(7.t) = 0.

Answer 1

utt t ut unn OLN L32 3 19 PDE utt d = -x 2 x 80 tro ure, o)=1 ut(x,od=0 Ua(2,0-0 Unn, td=0 By method of sebarcation of variables let ce (ait)= X (2) TH) the sokution of given ut= X(0) T 'Ct) X(2) T"(t) unn="(n)TA) 0 = 4x (0,0) = x '10) T (L) It gives X '10)=0 and T(Lito ut similartes, X'(n)=0 putting 2, ③ & G But only tour do, we have required solution we have + d 7 3T let

CE TI + -२२ T x 3T X 22 on soking x(n)=c, los ant lasinda x' l- & [-ci sinun & calosta] o= x 'lo) - ula - (2=0 0 = X(R) = a[-ci sin an] But uto, Becaus otherwise zero solution sinar- O= sinon, ex dnan Xncnd- Cocos nn. Now 3+" +87 +327 0 O e) 3T" +T' +327 auxippars equation is 3 m2 +m+32 2 - 0

CE TI + -२२ T x 3T X 22 on soking x(n)=c, los ant lasinda x' l- & [-ci sinun & calosta] o= x 'lo) - ula - (2=0 0 = X(R) = a[-ci sin an] But uto, Becaus otherwise zero solution sinar- O= sinon, ex dnan Xncnd- Cocos nn. Now 3+" +87 +327 0 O e) 3T" +T' +327 auxippars equation is 3 m2 +m+32 2 - 0

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