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how do I solve this differential equation $f(x+1)=f'(x)$?

Background:
I have not taken any collage courses or any calculus courses yet, I'm still in high school. I know a lot of math because of videos from 3blue1brown and other, and I do practice problems I make up. Today I was trying to solve many differential equations some I couldn't do and some I was able to do like $h'(x)=h(x)x^2$. first I right $h(x)$ in terms of $e^{g(x)}$ when I take the derivative of it and I get $g'(x) e^{g(x)}=x^2 e^{g(x)}$ then I get $g(x)$ to be $\frac{x^3}{3}$ so $h(x)= e^{\frac{x^3}{3}}$. I know how to solve some differential equations.

I would like to learn a new technique to solve differential equations when the input changes. I would like to know the answer to this differential equation. But I also want to learn how to solve more types of differential equations. Any tips would help.

User85825
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    If you write down a random differential equation, almost certainly you will not be able to solve it. –  Jul 11 '21 at 03:14
  • @MichaelBarz good to know. Is this one possible? – User85825 Jul 11 '21 at 03:15
  • I would be very surprised if it was. –  Jul 11 '21 at 03:16
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    Where did you find these problems? If they are from a book, then it's very likely that the techniques which are neccesary to solve them are somewhere in the pages before the problems. – Jackozee Hakkiuz Jul 11 '21 at 03:16
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    Technically, that's not a differential equation. It's a differential-difference equation, also called a delay-differential equation, and the solution techniques are different, and generally not taught in the undergrad math curriculum. – Gerry Myerson Jul 11 '21 at 03:19
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    Such equations come up in analytic number theory. See, e.g., https://en.wikipedia.org/wiki/Dickman_function – Gerry Myerson Jul 11 '21 at 03:21
  • @JackozeeHakkiuz I tried to make up a seemingly simple problem, I don't have any math books at home, my family hates math. – User85825 Jul 11 '21 at 03:21
  • Your family hates maths? See if you can test into a gifted program: if you want to discuss this further, please do so in chat. – Toby Mak Jul 11 '21 at 06:35
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    See also https://math.stackexchange.com/questions/199691/find-f-where-fx-f1x. – Hans Lundmark Jul 11 '21 at 07:38
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    Notice that there are interesting solutions for $f(x+1)=Cf'(x)$ for some constant $C$, such as $f(x)=\sin\left(\frac{\pi}{2}x\right)$. – John Wayland Bales Jul 11 '21 at 16:01

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