The authors consider a spectral type linear differential equation with polynomial coefficients of order \(N(\geq 1)\), namely: \[ L_N[y](x)= \sum_{i=1}^N \sum_{j=0}^i \ell_{ij}x^jy^{(i)}(x)= \lambda_ny(x), \tag{\(*\)} \] where \(\ell_{ij}\) are given constants while \[ \lambda_n= n\ell_{11}+ n<(n-1)\ell_{22}+\dots+ n(n-1)\dots (n-N+1)\ell_{NN}, \] and show that if an orthogonal polynomial system \(\{P_n(x) \}_{n=0}^\infty\) relative to a regular moment functional \(\sigma\) satisfies the differential equation \((*)\), then it consists of the classical Hermite polynomials (after suitable linear change of variable) if and only if \(\ell_N(x)\equiv \sum_{j=0}^N \ell_{Nj}x^j\) is a constant.