TITLE: Function that is the sum of all of its derivatives
QUESTION [51 upvotes]: 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:
$$f(x)=\sum_{n=1}^\infty f^{(n)}(x)$$

REPLY [4 votes]: the question means:
$$y-y'-y''-y'''-......y^n=0$$
the differential equation is homogeneous and the characteristics equation is
$$1-r-r^2-r^3-.......r^n=0$$
$$r(1+r+r^2+r^3+....)=1$$
by using the geometric series (r<1)
$$\frac{r}{1-r}=1$$
$$r=1-r$$
$$r=\frac{1}{2}$$ 
so the function is
$$y=Ce^{\frac{x}{2}}$$