  In this work, we consider rational ordinary differential equations dy/dx =
Q(x,y)/P(x,y), with Q(x,y) and P(x,y) coprime polynomials with real
coefficients. We give a method to construct equations of this type for which a
first integral can be expressed from two independent solutions of a
second-order homogeneous linear differential equation. This first integral is,
in general, given by a non Liouvillian function. We show that all the known
families of quadratic systems with an irreducible invariant algebraic curve of
arbitrarily high degree and without a rational first integral can be
constructed by using this method. We also present a new example of this kind of
families. We give an analogous method for constructing rational equations but
by means of a linear differential equation of first order.
