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  "body": "\n## Summary\n\nDifferential Galois theory studies linear differential equations over a [[differential field|GA5PTQGC6G]], see the [[definition|RW8DQQYJMR]].\nIt uses the group of symmetries of their solutions, as [[Galois theory|WNQDKQYZ2R]] does for polynomial equations.\nThe solutions generate a [[Picard–Vessiot extension|TB53Y5NMJD]], and the differential Galois group is the group of its automorphisms that commute with the derivation.\nThis group is a linear algebraic group over the constants, and there is a Galois correspondence between its closed subgroups and the intermediate differential fields.\nThe theory explains why some equations, and some integrals such as $\\int e^{-x^2} \\, dx$, cannot be solved in elementary terms, in line with [[Liouville's theorem|8VRTPECBPZ]].\n\n## Explanations\n\n### L1 human\n\nClassical Galois theory explains why the quintic equation has no formula in radicals, by looking at the symmetries of its roots.\nDifferential Galois theory does the same for linear differential equations such as $y'' + x y = 0$.\nThe symmetries now form a group of matrices that can be continuous, such as all rotations or all scalings.\nFor $y' = y$ over the rational functions, the solutions are $c e^x$, and the symmetries are the scalings $e^x \\mapsto c e^x$ with $c \\ne 0$.\nWhether an equation can be solved by integrals, exponentials and algebraic functions depends on whether this group is solvable in a suitable sense.\n",
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