My question can be most simply put as how can we guarantee that the only solutions of a homogenous linear differential equation are of the form ce^at?(adsbygoogle = window.adsbygoogle || []).push({});

or for that matter that the particular solution of a function is given by u(x)f(x)

in the variation of parameters method.

how can we guarentee that there aren't other functions that also meet the requirements of the differential equation?

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# Assumptions of solution form

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