Computing divisors and common multiples of quasi-linear ordinary differential equations

If solutions of a non-linear differential equation may be obtained by specialization of 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 quasi-linear common multiple of a pair of quasi-linear equations always exists and design an algorithm which yields a quasi-linear common multiple.