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3 types of PDE
From the general form of the linear homogeneous PDE, we can divide all PDE into 3 types: 1.

$\Delta\equiv B^{2}-4AC>0\Rightarrow$ Hyperbolic PDE(Wave Equation):

$\phi_{tt}=c\phi_{xx}$ 2.

$\Delta\equiv B^{2}-4AC=0\Rightarrow$ Parabolic PDE(Heat/Diffusion Equation):

$\phi_{t}=k\phi_{xx}$ 3.

$\Delta\equiv B^{2}-4AC<0\Rightarrow$ Elliptical PDE(Laplace Equation):

$\phi_{xx}+c\phi_{yy}=0$

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