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  "body": "\n## Definition\n\nThe *Tricomi equation* is the [[partial differential equation|GC5BDZ6QBC]]\n$$y \\, u_{xx} + u_{yy} = 0.$$\nThe unknown is a real function $u(x, y)$ on an open subset of $\\R^2$.\nSubscripts denote [[partial derivatives|09EGC6F8PT]].\n\n## Remarks\n\nThe equation is [[elliptic|B4TTNHSTRX]] where $y \u003e 0$, [[hyperbolic|92C3S03AQZ]] where $y \u003c 0$, and parabolic on the line $y = 0$, so it is of [[mixed type|7WFKPAMQFG]].\nIn the half-plane $y \u003c 0$ its characteristic curves are $3x \\pm 2(-y)^{3/2} = \\text{const}$.\nSome authors write it as $u_{xx} = x \\, u_{yy}$, which is the same equation after renaming and reflecting the variables.\n\nExample: $u(x, y) = x^2 - \\tfrac{1}{3} y^3$ is a solution, because $y \\, u_{xx} + u_{yy} = 2y - 2y = 0$.\n\nNon-example: $u(x, y) = x^2 + y^2$ is not a solution, because $y \\, u_{xx} + u_{yy} = 2y + 2$ is not identically $0$.\n",
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