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3 types of linear homogeneous 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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