infinitesimal generators of lie symmetry group of parametric ordinary differential equations

Lie’s theory of symmetry groups plays an important role in analyzing and solving differential equations; for instance, by decreasing the order of equation. moreover, there are some analytic methods to find the infinitesimal generators that span the lie algebra of symmetries. in this paper, we first converted the problem of finding infinitesimal generators in to the problem of solving a system of polynomial equations in the context of computational algebraic geometry. then, we used gröbner basis a novel computational tool to solve this problem. as far as we know, when a differential equation contains some parameters, there is no linear algebraic algorithm up to our knowledge to deal with these parameters; so, we must apply the algorithms, which are based on gröbner basis.