  The Podolsky generalized electrodynamics with higher derivatives is
formulated in the first-order formalism. The first-order relativistic wave
equation in the 20-dimensional matrix form is derived. We prove that the
matrices of the equation obey the Petiau-Duffin-Kemmer algebra. The
Hermitianizing matrix and Lagrangian in the first-order formalism are given.
The projection operators extracting solutions of field equations for states
with definite energy-momentum and spin projections are obtained, and we find
the density matrix for the massive state. The $13\times 13$-matrix Schrodinger
form of the equation is derived, and the Hamiltonian is obtained. Projection
operators extracting the physical eigenvalues of the Hamiltonian are found.
