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what is \(L\) ?

Linear DE operated on...

I know that it's obvious, but I've yet to encounter a formal proof.

i can not see the differential equation

Oh, it's just the general sign for\[Ly=\sum_{i=0}^{i=n}a_i\frac{d^i}{dx^i}y=0\]

Or is this only provable for more specific DE?

i thought there were usually as many constants in the solution as the order of the DE

Yes, but they're all in the complimentary solution.

I've not come across 'roots' with regards to DE, but I suppose so, yes.

actually the roots are in the complementary solution