Inspired by L'Hosopital I started considering “ L'Hospital's equations ”
Let $x$ be real and for all $ x > s $ we have $f(x),g(x)$ map to the reals. Let $*^{(n)} $ denote the $n$ th derivative. $n$ is a strict positive integer.
Then L'Hospital's equations are
$$ \frac{f^{(n)}(x)}{g^{(n)}(x)} = H\Bigl(\frac{f(x)}{g(x)}\Bigr) $$
Where $H(x) $ is a function.
Many solutions are probably known and I assume there is a general method to find solutions to them.
Uniqueness , existence etc are interesting too.
The equations can be connected.
As an example
$$H(x) = \frac{1}{x} $$
The a solution to
$$ \frac{f’(x)}{g’(x)} = H\Bigl(\frac{f(x)}{g(x)}\Bigr) $$
Is also a solution to
$$ \frac{f’’(x)}{g’’(x)} = \frac{f(x)}{g(x)} $$
And one wonders about the uniqueness of them , In fact one wonders If
$$ \frac{f’(x)}{g’(x)} = H^*(\frac{f(x)}{g(x)}) $$
With
$$ H^*(H^*(x)) = x $$
Has different solutions then the two above ?
I assume understanding all this would make nice examples of limit exercises using l’hopital. Or nice examples of solving differential equations. However it is unclear to me If this adds value to limits or differential equations. Ofcourse my lack of knowledge about them and What is known about them is the main reason for that.
I would like to point out that
$$ \frac{f’(x)}{g’(x)} = H(\frac{f(x)}{g(x)}) $$ With $H(x) = 1/x $ has solutions that do NOT satisfy
$$ f’(x) = \frac{1}{f(x)} $$
And such a “ non-trivial “ solution is known.
Lastly , not all differential equations have a solution.
Example $ f'(x) = \exp(g(x)) , g(f(x)) = f(g(x)) = x.$
I searched for these online but I could not find these “ l’hopital equations ”.
In fact I rarely see differential equations with 2 functions involved where both are of 1 variable , but maybe I overlooked. And Maybe most do not have a closed form.
Nevertheless , I want to know the solutions that have a closed form , or how to find them. And to understand the ones that do not too !
Are taylor series useful here ?