Question

1. Solve the following IBVPs for a string of unit length, subject to the given conditions 100, t) = 0, u(1,t)-0, t>0 a) b) a(x, 0) = f(x), ut (x,0) = g(x) 0 < x < 1 f(x) = sinocoso, g(x) = 0, c = 1/л. f(x) = sinx + (1/2)sin3m + 3sin7m, g(x) = snLTX, C = 1 0, if 0sxs 30( 3 30

Answer 1

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"Let u(x_1 + 1) = X(x) T(t) so, T""/c^2 T = X""/X = - lambda^2 T"" + lambda^2 C^2 T = 0, X"" + lambda^2 X = 0 T(t) = c_1 cos (lambda c t) + c_2 sin (lambda c t), X(x) = c_3 cos lambda x + c_4 sin lambda x X(0) = 0 c_3 = 0 X(1) = 0 lambda = n pi So, u(x_1 + 1) = sigma^infinity_n = 1[A_n cos (cn pi t) + B_n sin (nc pi t)] sin (n pi x) a) u(x, 0) = sin pi x cos pi x = sin(2 pi x)/2 sigma A_n sin (n pi x) = sin(2 pi x)/2 A_2 = 1/2, other A_n = 0 u_t(x, 0) = 0 B_n = 0 C = 1/pi So, u(x, t) = 1/2 cos(t) sin (2 pi x) \r\n u(x, 0) = sin(pi x) + 1/2 sin (3"

In last equation of last picture. , An and Bn are define above in last picture in closed bracket..