English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

This paper is devoted to a complete parametric study of Liouvillian solutionsof the general trace-free second order differential equation with a Laurentpolynomial coefficient. This family of equations, for fixed orders at $0$ and$\infty$ of the Laurent polynomial, is seen as an affine algebraic variety. Weproof that the set of Picard-Vessiot integrable differential equations in thefamily is an enumerable union of algebraic subvarieties. We compute explicitlythe algebraic equations of its components. We give some applications to wellknown subfamilies as the doubly confluent and biconfluent Heun equations, andto the theory of algebraically solvable potentials of Shr\"odinger equations.Also, as an auxiliary tool, we improve a previously known criterium for secondorder linear differential equations to admit a polynomial solution.

Liouvillian solutions for second order linear differential equations with Laurent polynomial coefficient