A sequence is holonomic if its terms obey a linear recurrence relation with polynomial coefficients. In this paper we consider the Positivity Problem for second-order holonomic sequences with linear coefficients, i.e., the question of determining, for a given sequence $\langle u_n\rangle_n$ obeying the recurrence $(a_1n+a_0)u_n=(b_1n+b_0)u_{n−1}+(c_1n+c_0)u_{n−2}$, whether all terms of $\langle u_n\rangle_n$ are non-negative. Our main result is to establish decidability in case the characteristic roots of the recurrence are rational and distinct. We achieve this by leveraging recent results on effective transcendence of values of E-functions and 1-periods, which are integrals playing a central role in the theory of algebraic curves.