Inspired by the paper of Bonichon, Bousquet-M\'elou, Dorbec and Pennarun, we give a system of functional equations for the ogfs for the number of planar Eulerian orientations counted by the edges, U(x), and the number of 4-valent planar Eulerian orientations counted by the number of vertices, A(x). The latter problem is equivalent to the 6-vertex problem on a random lattice, widely studied in mathematical physics. While unable to solve these functional equations, they immediately provide polynomial-time algorithms for the coefficients of the generating function. From these algorithms we have obtained 100 terms for U(x) and 90 terms for A(x). Analysis of these series suggests that they both behave as ⋅(1−μx)2/log(1−μx), where we make the confident conjectures that μ=4π for Eulerian orientations counted by edges and μ=43–√π for 4-valent Eulerian orientations counted by vertices.