Via the technique of constructing jet differentials through holomorphic Morse inequalities, we give a partial answer to the Green-Griffiths conjecture in a very wide context : for every projective variety of general type X, there exists a global algebraic differential operator P on X (in fact many such operators Pj) such that every entire curve must satisfy the differential equations . We recover from there the result of Diverio-Merker-Rousseau (2009) confirming the Green-Griffiths conjecture for generic hypersurfaces of high degree in projective space (with an even better bound asymptotically as the dimension tends to infinity), as well as a recent recent of Diverio-Trapani (2010) on the hyperbolicity of generic 3-dimensional hypersurfaces in P4; the latter results also depend on a use of Siu's differentiation technique via meromorphic vector fields, as improved by Pǎun and Merker.