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  "body": "\n## Summary\n\nKovacic'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.\nIt works on the reduced form $y'' = r(x)\\, y$ with $r \\in \\C(x)$.\nIt is based on the classification of the algebraic subgroups of $\\mathrm{SL}_2(\\C)$, the possible [[differential Galois groups|EEZS4H8QPH]] of such equations.\nIt is implemented in computer algebra systems and is the standard tool for closed-form solution of these equations.\nSee the [[definition|MMEX2JZWGR]].\n\n## Explanations\n\n### L1 human\n\nGiven an equation like $y'' = (x^2 + 1)\\, y$, one wants to know whether it has a solution built from exponentials, integrals and algebraic functions.\nHere the answer is yes: $y = e^{x^2/2}$ works, since $y'' = (x^2 + 1)\\, e^{x^2/2}$.\nKovacic's algorithm answers this question in finitely many steps.\nIt tries a short list of possible shapes for a solution, one for each kind of symmetry group the equation could have.\nIf no shape fits, the equation has no closed-form solution of this kind, as for the Airy equation $y'' = x y$.\n",
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