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  "body": "\n## Summary\n\nLinear second-order partial differential equations are sorted into elliptic, parabolic and hyperbolic equations.\nThe type is given by the signs of the eigenvalues of the matrix of second-order coefficients.\nThe three types behave very differently: elliptic equations describe equilibria, parabolic equations describe diffusion, and hyperbolic equations describe waves.\nThe model cases are the [[Laplace equation|S8RG6N1PRT]], the [[heat equation|VP2HN86VAG]] and the [[wave equation|5EHD2XBQDG]].\nSee the [[definition|J53A2A04P7]].\n\n## Explanations\n\n### L1 human\n\nIn two variables the second-order part $A u_{xx} + 2B u_{xy} + C u_{yy}$ looks like the quadratic form $A x^2 + 2B xy + C y^2$.\nSuch a form describes an ellipse, a parabola or a hyperbola, depending on the sign of $AC - B^2$, and the names of the PDE types come from this.\nFor $u_{xx} + u_{yy}$ one has $AC - B^2 = 1 \u003e 0$, an ellipse, and the equation is elliptic.\nFor $u_{xx} - u_{yy}$ one has $AC - B^2 = -1 \u003c 0$, a hyperbola, and the equation is hyperbolic.\nThe type tells which extra data, such as boundary values or initial values, give a good problem.\n",
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