For a particular set of homogeneous boundary conditions, and a particular type of root of theauxiliary equation (ie repeated, distinct real, or complex), the non-zero eigenvalues associatedwith the boundary value

problem for X are found to satisfy the equation \mu=-\tanh (\mu L), \quad \mu \neq 0 Thus, for this case of eigenvalue, there are+ solutions to theODE because the above equation has