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  "body": "\n## Definition\n\nLet $r \\in \\C(x)$ be a rational function and consider the equation $y'' = r\\, y$ over the differential field $\\C(x)$.\nLet $G$ be its [[differential Galois group|RW8DQQYJMR]], the group of differential automorphisms of a [[Picard–Vessiot extension|H5P06V330V]].\nThen $G$ is an algebraic subgroup of $\\mathrm{SL}_2(\\C)$, and exactly one of the following holds.\n\n1. $G$ is conjugate to a group of upper triangular matrices; then there is a solution $y = e^{\\int \\omega}$ with $\\omega \\in \\C(x)$.\n2. Case 1 fails and $G$ is conjugate to a group of matrices that are diagonal or anti-diagonal. Then some solution is $y = e^{\\int \\omega}$ with $\\omega$ algebraic of degree $2$ over $\\C(x)$.\n3. Cases 1 and 2 fail and $G$ is finite. Then all solutions are algebraic, and some solution is $y = e^{\\int \\omega}$ with $\\omega$ algebraic of degree $4$, $6$ or $12$.\n4. $G = \\mathrm{SL}_2(\\C)$; then the equation has no nonzero [[Liouvillian|R80FA7993W]] solution.\n\n*Kovacic's algorithm* tests cases 1, 2 and 3 in turn.\nIn each case it builds finitely many candidate exponents from the expansions of $r$ at its poles and at infinity.\nThen it searches for $\\omega$ by solving for a polynomial of bounded degree.\nIt returns a Liouvillian solution, or the answer that none exists.\n\n## Remarks\n\nEvery equation $y'' + a(x)\\, y' + b(x)\\, y = 0$ with $a, b \\in \\C(x)$ can be brought to the reduced form: $z = y\\, e^{\\frac12 \\int a}$ satisfies $z'' = r z$ with $r = \\tfrac14 a^2 + \\tfrac12 a' - b$.\nExample: for $r = 1$, the solution $y = e^{x}$ has $\\omega = y'/y = 1 \\in \\C(x)$, so the equation falls in case 1.\nNon-example: for the Airy equation, $r = x$, the algorithm finds no candidate in cases 1 to 3, so $G = \\mathrm{SL}_2(\\C)$ and no nonzero solution is Liouvillian.\nIn case 3 the group modulo $\\pm 1$ is the tetrahedral, octahedral or icosahedral group, and these correspond to the degrees $4$, $6$ and $12$.\nJerald Kovacic published the algorithm in 1986.\n",
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