Question

Show that sin (kx) and cos (kx) given in Eq. (11-48a) are two independent solutions of the differentia equation, Eq. (11-47a). Consider a rectangular wavequide haing dimoneinc 404

Answer 1

Here I'm using wronskian method to show solution are independent answer is below. Thank you

Given differential equation is d^2 X(x)/dx^2 + k_x^2 = 0 -----(1) It is homogeneous differential equation Therefore D^2 X(x) + k_x^2 X(x) = 0 ---(D = d X(X)/dx) Therefore (D^2 + k_x^2) X(x) = 0 Auxiliary equation is m^2 + k_x^2 = 0 Therefore m^2 = -k_x Therefore m = plusminus k_x \r\n Therefore general solution of (1) is X(x) = A cos(k_x x) + B sin(k_x x) or X(x) = A sin (k_x x) + B cos(k_x x) let y_1 = sin(k_x x), y_2 = cos(k_x x) be soln of (1) we show that they are independent We find Wronskian Therefore W (y_1, y_2) = vert y_1 dy_1/dx y_2 dy_2/dx vert = vert sin(k_x x) k_x cos(k_x x) cos(k_x x) -k_x sin k_x vert = -k_x sin^2(k_x x)- k_x cos^2 (k_x x) = -k_x (sin^2 (k_x x) + cos^2 (k_x x)) = -k_x x (1) ---(sin^2 x + cos^2 x = 1) = -k_x notequalto 0