ALGORITHMS FOR SIMPLIFYING DIFFERENTIAL EQUATIONS

Abstract

We present three algorithms to reduce homogeneous linear differential equations to their simplest form. Factoring a differential operator reduces a differential equation L(y) = 0 to equations of minimal order, but this is not the only simplification one can make. There are three order-preserving transformations that can change the degree of the coefficients. To fully simplify a differential equation, after the order is minimized, we want to find the smallest equation that can be reached under any order-preserving transformations. We design algorithms to find transformations that reduce L to its simplest form under all three transformations. The algorithms use relative invariants and integral bases for differential operators. We give an algorithm to find all relative invariants, and we generalize a prior integral basis algorithm to cover all cases.