I was wondering how the general formula for the solutions of an nth order linear homogeneous ODE that had a characteristic equation which could be factored to (x-a)^n was derived(IE a set of solutions consisting e^(mx), x*e^(mx), ....x^(n-1)e^(mx)))?(adsbygoogle = window.adsbygoogle || []).push({});

For example the ODE,

y^(3)- 3y'' + 3y' - y = 0,

with characteristic equation,

m^3 - 3m^2 + 3m - 1 = 0

can be factored to

(m-1)^3,

where m = 1

and e^(x), x*e^(x) and x^2*e^(x) are all solutions.

For a second order ODE this can be found using a reduction of order technique but for higher order ODE's it gets very difficult to do so I am wondering what proof/explanation exists to show that we know such solutions exist?

I have looked around online and all the books/articles just say thats the case but don't provide an explanation.

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# Question on ODE's with roots of multiplicity.

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