---
tag: XC4245C9E5
kind: concept
name: Kovacic's algorithm
msc: [34M15, 12H05, 68W30]
sources:
  - url: https://en.wikipedia.org/wiki/Picard%E2%80%93Vessiot_theory
---

## Summary

Kovacic's algorithm decides whether a second-order linear differential equation with rational function coefficients has a [[Liouvillian|N5PM2NCPWF]] solution, and finds one when it exists.
It works on the reduced form $y'' = r(x)\, y$ with $r \in \C(x)$.
It is based on the classification of the algebraic subgroups of $\mathrm{SL}_2(\C)$, the possible [[differential Galois groups|EEZS4H8QPH]] of such equations.
It is implemented in computer algebra systems and is the standard tool for closed-form solution of these equations.
See the [[definition|MMEX2JZWGR]].

## Explanations

### L1 human

Given an equation like $y'' = (x^2 + 1)\, y$, one wants to know whether it has a solution built from exponentials, integrals and algebraic functions.
Here the answer is yes: $y = e^{x^2/2}$ works, since $y'' = (x^2 + 1)\, e^{x^2/2}$.
Kovacic's algorithm answers this question in finitely many steps.
It tries a short list of possible shapes for a solution, one for each kind of symmetry group the equation could have.
If no shape fits, the equation has no closed-form solution of this kind, as for the Airy equation $y'' = x y$.