I way toying around with some functions and came up with the following sequence.

$$ 0,1,x+1,e^x-1,-\ln(e^{x+1}+1),... $$

If $y_n=f_n(x)$, then $y_{n+1}$ is obtained using $$ \frac{dy_{n+1}}{dx} = f_n(y_{n+1})$$

The sequence above was derived using $y_0=0$, and by setting all subsequent constants of integration to 1. I wasn't able to go any farther than that, since the next term proved too difficult for me to calculate.

Here's my question: Can you find a $y_0$ and a value for the constants of integration such that this sequence contains a finite number of distinct functions?

I found a trivial set of solutions, $y_0 = ax+b$ with constants of integration set to zero. This yields

$$ ax+b, -\frac{b}{a}, -\frac{bx}{a}, 0, 0,... $$ I'm curious to know if there are any others. In particular, are the any periodic sequences of this form? (with period greater than one, preferably.)


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