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I try to have a general expression of a sequence of functions defined as follow :

$$u_0=f ~\text{and}~ \forall n >0, u_{n+1} =-f\times u_n + \frac{du_n}{dt} $$

where $f\in \mathcal{C}^\infty(\mathbb{R},\mathbb{R}) $. Does anyone know how to compute the expression of $u_n$, a method or a problem related ?

I already try to figure out a expression based on the first terms but I struggle to generalize it...

Thank you very much in advance !

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If $g'=-fg$ and $v_n=u_ng$ then $v_n=v_0^{(n)}.$

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  • $\begingroup$ Thank you very much, so we can take $g=e^{-F} $ with $F$ a primitive of $f$ and it's done. Thank you Anne ! $\endgroup$
    – NancyBoy
    Mar 3, 2023 at 22:54
  • $\begingroup$ Yes this is what I meant. But don't thank by a comment: math.stackexchange.com/help/someone-answers $\endgroup$ Mar 3, 2023 at 23:22

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