Function that is the sum of all of its derivatives

I have just started learning about differential equations, as a result I started to think about this question but couldn’t get anywhere. So I googled and wasn’t able to find any particularly helpful results. I am more interested in the reason or method rather than the actual answer. Also I do not know if there even is a solution to this but if there isn’t I am just as interested to hear why not.

Is there a solution to the differential equation:

f(x)=n=1f(n)(x)

Answer

f(x)=exp(12x) is such a function, since f(n)=2nf(x), you have

n=1f(n)(x)=n=12nf(x)=(21)f(x)=f(x)

This is the only function (up to a constant prefactor) for which nf(n) and its derivatives converge uniformly (on compacta), as f=n=1f(n+1)=ff follows from this assumption. But this is the same as f2f=0, of which the only (real) solutions are f(x)=Cexpx2 for some CR.

Attribution
Source : Link , Question Author : o.comp , Answer Author : s.harp

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