The orthogonality of eigenfunctions in problems of unsteady heat conduction in an infinite slab with symmetric and nonsymmetric convective boundary conditions are demonstrated by performing actual integration of the products of the derived forms of eigenfunctions and by implementing the corresponding eigenvalue conditions. The analysis also applies to longitudinal vibrations of an elastic rod attached at its ends to linear elastic springs, and to advection-diffusion problems under appropriate boundary conditions. The same form of eigenfunctions and the same type of eigenvalue condition apply in all three considered cases. The proofs are compared with the well-known general proof of orthogonality of eigenfunctions, which circumvents the actual integration. The presented analysis is pedagogically appealing for use in engineering and applied physics education.