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  "body": "\n## Summary\n\nThe Lie symmetry method solves differential equations by using their continuous symmetries.\nA Lie point symmetry is a one-parameter group of changes of variables that maps solutions to solutions.\nIn adapted coordinates, called canonical coordinates, the symmetry becomes a translation and the equation can be integrated by quadrature.\nThe method, due to Sophus Lie, explains most of the classical tricks for first-order equations, such as the substitution $y = x v$ for a [[homogeneous first-order equation|0RN88KZBJ3]].\nSee the [[definition|WZKRG7H3FX]].\n\n## Explanations\n\n### L1 human\n\nMany solving tricks for differential equations look unrelated, but most of them use a hidden symmetry.\nFor example, the equation $y' = F(y/x)$ does not change if one scales $x$ and $y$ by the same factor.\nSo scaling a solution curve gives another solution curve.\nIn coordinates in which the scaling is a shift, such as $r = y/x$ and $s = \\ln\\abs{x}$, the equation no longer depends on $s$ and can be integrated directly.\n",
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