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  "body": "\n## Definition\n\nLet $U \\subseteq \\R^2$ be open and let $\\omega : U \\to \\R$ be smooth; consider the [[ordinary differential equation|0P97V20RW7]] $y' = \\omega(x, y)$.\nLet $\\Gamma_\\varepsilon : (x, y) \\mapsto (\\hat x, \\hat y)$ be a smooth local one-parameter group of maps, with $\\Gamma_0 = \\id$ and $\\Gamma_\\varepsilon \\circ \\Gamma_\\delta = \\Gamma_{\\varepsilon + \\delta}$.\nIt is a *Lie point symmetry* of the equation if it maps graphs of solutions to graphs of solutions.\nIts infinitesimal generator is the vector field $X = \\xi \\partial_x + \\eta \\partial_y$ with $\\xi = \\partial \\hat x / \\partial \\varepsilon$ and $\\eta = \\partial \\hat y / \\partial \\varepsilon$ at $\\varepsilon = 0$.\nThe group is a symmetry if and only if the linearized symmetry condition holds:\n$$\\eta_x + (\\eta_y - \\xi_x)\\, \\omega - \\xi_y\\, \\omega^2 = \\xi\\, \\omega_x + \\eta\\, \\omega_y,$$\nwhere subscripts denote [[partial derivatives|05AXG3QRKR]].\nThe *Lie symmetry method* solves the equation from a known symmetry, near a point where $X \\ne 0$, as follows.\n\n1. Find canonical coordinates $(r, s)$, functions of $(x, y)$ with $X r = 0$ and $X s = 1$.\n2. In these coordinates the equation takes the form $ds/dr = \\Omega(r)$, with no dependence on $s$.\n3. Integrate: $s = \\int \\Omega(r)\\, dr + C$, then return to $(x, y)$.\n\n## Remarks\n\nExample: for $\\omega(x, y) = F(y/x)$, the scalings $(x, y) \\mapsto (e^\\varepsilon x, e^\\varepsilon y)$ have generator $X = x \\partial_x + y \\partial_y$.\nCheck of the condition: the left side is $0 + (1 - 1)\\omega - 0 = 0$, and the right side is $x \\cdot (-y/x^2) F' + y \\cdot (1/x) F' = 0$.\nCanonical coordinates are $r = y/x$ and $s = \\ln\\abs{x}$, since $X r = -y/x + y/x = 0$ and $X s = 1$.\nThen $ds/dr = 1/(F(r) - r)$, and the equation is solved by an [[antiderivative|0BW9N5CDSB]].\nNon-example: the translations $(x, y) \\mapsto (x, y + \\varepsilon)$ are not symmetries of $y' = x y$, since $\\xi = 0$, $\\eta = 1$ give left side $0$ and right side $\\omega_y = x \\ne 0$.\nFor equations of order $n$, a symmetry reduces the order by one.\n",
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