The paper is devoted to the analysis of the TE-wave propagation in a planar waveguide with variable permittivity. The setting is reduced to a transmission eigenvalue problem for an ordinary differential equation where the spectral parameter is the wave propagation constant. For the computation of eigenvalues the shooting method is applied in the form specifically developed for this class of problems. The method allows one to calculate propagating, evanescent, and complex surface and leaky waves. A variety of numerical results illustrating the method is presented.