The analytical treatment of transient heat conduction problems for one- dimensional multilayered composites by the orthogonal expansion technique requires the solution of a corresponding eigenvalue problem. Such an eigenvalue problem is not of the conventional Sturm-Liouville type because of the discontinuities of the coefficient functions. An analytical solution of one transient heat conduction problem in one- dimensional multilayered slabs, cylinders and spheres is presented, which implements a safe algorithm for the automatic computation of the eigenvalues and the eigenfunctions of the resulting Sturm-Liouville type system.