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## Main Question or Discussion Point

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?

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?

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?