Question

Previous Problem Problem List Ned Probiem (1 point) Use generalized Fourier series to solve the IEVP for the wave equation 87u lnpartatar wit, μ.-(2n-1)/2 we seek a soluton inthe torn where a, and bn are generalized Fourier coefficients ns prociem You have urlimted stemots

Answer 1

Answer are in closed bracket in 2nd image..

Any problem in any answer then comment below.. i will help you.

u_tt = u_xx let u (x, t) = x (x) T (t) T''/T = x''/x = -lambda^2 T'' + lambda^2 T = 0, x'' + lambda^2 x = 0 T (t) = C_1 cos lambda t + c_2 sin lambda t x (x) = c_3 cos lambda x + c_2 sin lambda x T' (0) = 0 x (0) = 0 rightarrow c_3 = 0 rightarrow C_2 = 0 x' (pi) = 0 rightarrow cos (lambda pi) = 0 lambda = (2 n - 1)/2 u (x, t) = sigma An cos ((2n - 1)/2 t) sin ((2n - 1) x/2) u (x, 0) = x rightarrow sigma An sin ((2n - 1)x/2) = x An = 2/pi integral_0^pi x sin ((2n - 1) x/2) dx = 2/pi [-x cos (2n - 1)/2 x)/(2n - 1)/2 + sin ((2n - 1) x/2)/((2n - 1/2)^2]_0^pi \r\n = 2/pi [0 + sin ((2n - 1 pi/2)/(2n - 1)^2/4] = 8/pi (2 - 1)^2 (-1)^n - 1