Orthogonal polynomials with weight exp[$−NV (x)$] are studied where $V (x) = \sum_{k=1}^{d} a_{2k} x^{2k}$ is a polynomial of order $2d$. The generalized Freud’s equations for $d = 3, 4$ and 5 are derived and using this $R_{\mu} = h_{\mu} / h{\mu−1}$ is obtained, where $h_{\mu}$ is the normalization constant for the corresponding orthogonal polynomials. Moments of the density functions, expressed in terms of $R_{\mu}$ , are obtained using Freud’s equation and using this, explicit results of level densities as $N \rightarrow \infty$ are derived using the method of resolvents. The results are compared with those using Dyson–Mehta method.