On 7/26/2012 4:51 AM, Invisible wrote:
>
> Now I can't help but wonder... Is this a coincidence? Or some deep
> fundamental result? Is there some way of writing a solution to an
> equation like

Everything you've mentioned including the sinh and cosh stuff is a 
pretty immediate consequence of how e^(a*x) differentiates and the fact 
that solutions to the ODE's you're looking at are closed under linear 
combinations.  Think in terms of roots of unity and you should see how 
it can easily be extended to nth derivatives.

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