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What are the possible known methods for solving first order PDEs in higher dimensions? Is there anything else besides the method of characteristic curves?

In particular, I have four first order, linear, non-homogeneous PDEs in 8 dimensions which I (unfortunately) have to solve analytically.

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    $\begingroup$ Characteristic curves are the way to go, in general. $\endgroup$ – 5xum Feb 9 '16 at 10:11

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