In this paper we investigate (d + 1)-point D-optimal designs for d-th degree polynomial regression with weight function w(x) > 0 on the interval [a, b]. Suppose that w'(x)/w(x) is a rational function and the information of whether the optimal support contains the boundary points a and b is available. Then the problem of constructing (d + 1)-point D-optimal designs can be transformed into a differential equation problem leading us to a certain matrix with k auxiliary unknown constants. We characterize the weight functions corresponding to the cases when k= 0 and k= 1. Then, we can solve (d + 1)-point D-optimal designs directly from differential equation (k = 0) or via eigenvalue problems (k = 1). The numerical results show us an interesting relationship between optimal designs and ordered eigenvalues.