Solving Linear Differential Problems with Parameters

Abstract

We present algorithms for parametric problems in differential algebra that can be formulated in a suitable first-order language L. The atomic L-formulas are linear ODEs of arbitrary order with parametric coefficients of arbitrary degrees. Using rather weak axioms on differential fields or differential algebras that are realized in natural function domains, we establish explicit quantifier elimination algorithms for L providing also parametric sample solutions for purely existential problems. These sample solutions are “generic” solutions of univariate parametric linear ODEs that can be realized by concrete functions in the natural function domains mentioned above. We establish upper complexity bounds for the elimination algorithms that are elementary recursive for formulas of bounded quantifier alternation, in particular doubly exponential for existential formulas. Our results are in contrast to Seidenberg’s model theoretic elimination theory for non-linear problems that is non elementary recursive, requires very strong axioms that are not realizable in natural function domains, and does not provide sample solutions.