Question

Can someone expand on they got the eigenvalues and eigen funtctions? It looks like its a basic rule in ODE that I just forgot.

The left-hand side depends only on t and the right-hand side only on x,so they are both equal to some constant -1 The boundary conditions, ф(0)-0. dL)-0 make the x-dependent ordinary differential equation one of our standard problems, with eigenvalues λ for n=1,2,3, and corresponding Eigen functions ,-sin

Answer 1

1/phi partial differential^2 phi/partial differential x^2 = -lambda where phi is function of x then 1/phi (x) d^2 phi/dx^2 = -lambda 1/phi (x) d^2 phi/dx^2 = -mu^2 (let lambda = mu^2) - (1) d^2 phi/dx^2 = -mu^2 phi (x) partial differential^2 phi/partial differential x^2 + mu^2 phi (x) = 0 Auxiliary eq^n m^2 + mu^2 = 0 m = plusminus i mu phi (x) = c_1 cos mu x + c_2 sin mu x - (II) given boundary condition phi (0) = 0 phi (1) = 0 phi (0) = c_1 cos mu + c_2 sin mu = 0 = c_1 + 0 = c implies c_1 = 0 put c_1 = 0 in eq^n (1) phi (x) = c_2 sin mu x -(1) put phi (L) = 0 in eq^n (1) phi (L) = c_2 sin mu L = 0 \r\n c_2 sin mu L = 0 c_2 notequalto 0 implies sin mu L = 0 Implies sin mu L = sin n pi where n = 1, 2, 3... compare both sides