Given a matrix A in F(t)[D;\delta]^{n\times n} over the ring of differential polynomials, we first prove the existence of the Hermite form H of A over this ring. Then we determine degree bounds on U and H such that UA=H. Finally, based on the degree bounds on U and H, we compute the Hermite form H of A by reducing the problem to solving a linear system of equations over F(t). The algorithm requires a polynomial number of operations in F in terms of the input sizes: n, deg_{D} A, and deg_{t} A. When F=Q it requires time polynomial in the bit-length of the rational coefficients as well
