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The name is a bit obscure if you don't think in very generalized terms right out of the gate. It refers to a linear polynomial of multiple variables, which by definition must have the super simple form of:
and then we just put the unknown and each derivative into that simple polynomial:
except that now the are not just constants, but they can also depend on the argument (but not on or its derivatives).
Explicit solutions exist for the very specific cases of:


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