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One reason why the iteration

$$x_{n+1}=x_n-\tan\;x_n$$

converges quickly for appropriate starting values is that this is nothing more than the Newton-Raphson iteration for $\sin\;x$.

This got me thinking: given some arbitrary function $g(x)$, is there always a function $f(x)$ such that

$$\frac{f(x)}{f^\prime(x)}=g(x)$$

or are there restrictions on the nature of $g(x)$ so that the differential equation has a solution?

2 Answers 2