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I see the connection between the PDEs and the equations of conic sections, but why is that important?

I am under the impression that one of the big differences between the wave equation and the heat equation is that solutions to the wave equation are periodic in time (somehow conserving 'energy' or 'information' perhaps?) and the solutions to the heat equation diffuse and tend to a steady state (somehow losing energy or information). Is something like that generally true about hyperbolic and parabolic PDEs?

Are there other important motivations/explanations I should be aware of?

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It is important to understand that PDEs is a very broad field, even though it is easy to write down an equation for a general PDE. For example, complex analysis and ODEs could be seen as special cases of PDE-theory. As a consequence, some methods only work for certain PDEs. Mathematicians have grouped them together depending on what methods are tractable, so that we can have nice theorems that are valid for, say for example, elliptic PDEs.

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