The classical eigenvalue problems of mathematical physics are solved by means of a Laplace Transform extended, not over the interval (0, ∞) but over the interval of interest for the differential equation. The method is applied to the Hermite, Laguerre and Bessel equations and to the equation for the hypergeometric polynomials which include the Legendre, Tschebyscheff, and Jacobi polynomials as special cases.