Computing Divisors and Common Multiples of Quasi-linear Ordinary Differential Equations

If solutions of a non-linear differential equation are contained in solutions of another equation we say that the former equation is a generalized divisor of the latter one. We design an algorithm which finds first-order quasi-linear generalized divisors of a second-order quasi-linear ordinary differential equation. If solutions of an equation contain solutions of a pair of equations we say that the equation is a common multiple of the pair. We prove that a common multiple of a pair of quasi-linear equations always exists and design an algorithm which yields a common multiple. Introduction The problem of factoring linear ordinary differential operators L = T ◦ Q was studied in [10]. Algorithms for this problem were designed in [3], [11] (in [3] a complexity bound better than for the algorithm from [10] was established). In [5] an algorithm is exhibited for factoring a partial linear differential operator in two variables with a separable symbol. In [4] an algorithm is constructed for finding all the first-order factors of a partial linear differential operator in two variables. A generalization of factoring for D-modules (in other words, for systems of linear partial differential operators) was considered in [6]. A particular case of factoring for D-modules is the Laplace problem [2], [14] (a short exposition of the Laplace problem one can find in [7]). The meaning of factoring for search of solutions is that any solution of operator Q is a solution of operator L, thus factoring allows one to diminish the order of operators. Much less is known for factoring non-linear (even ordinary) differential equations. In Section 1 we design an algorithm for finding (first-order) generalized divisors of a secondorder quasi-linear differential equation. We note that our definition of generalized divisors is in the frames of differential ideals [9], rather than the definition of factorization from [13], [1] being in terms of a composition of nonlinear ordinary differential polynomials. In [1] a decomposition algorithm is designed.