What functions $f$ satisfy the equation $$f'(x)=\frac{f(\sqrt{1-x^2}) - f(x)}x$$
Could you kindly show how you arrived at the function?
$\begingroup$
$\endgroup$
11
-
$\begingroup$ Thanks for editing it. I didn’t really know how😅 $\endgroup$– David OkogbeninMar 21, 2020 at 19:05
-
1$\begingroup$ This might be help you in future. $\endgroup$– BumblebeeMar 21, 2020 at 19:11
-
1$\begingroup$ Hint: Note that $$f'(\sqrt{1-x^2})=-\dfrac{x}{\sqrt{1-x^2}}f'(x).$$ Use this to find $f'$ and then integrate. $\endgroup$– BumblebeeMar 21, 2020 at 19:15
-
$\begingroup$ Thanks again 🙏🏽 $\endgroup$– David OkogbeninMar 21, 2020 at 19:17
-
$\begingroup$ When u say f’((1-x^2)^1/2) are u referring to the derivative with respect to x or to (1-x^2)^1/2 $\endgroup$– David OkogbeninMar 21, 2020 at 21:20
|
Show 6 more comments
1 Answer
$\begingroup$
$\endgroup$
3
$$f'(x)=\frac{f(\sqrt{1-x^2}) - f(x)}x$$ $f(x)=g(x^2)$
$f'(x)=2xg'(x^2)$ $$2xg'(x^2)=\frac{g(1-x^2) - g(x^2)}{x}$$ $$2x^2g'(x^2)=g(1-x^2) - g(x^2)$$ $x^2=s$ $$2sg'(s)=g(1-s) - g(s)$$ $Dg(s)=g'(s)$
$Ag(s)=g(1-s)$ $$(2sD-A+1)g=0$$
$$\left(2sD-\exp\left(\pi isD\right)\exp(D)+1\right)g=0$$ Baker-Campbell-Hausdorff Formula
This solves $\ln(\exp(X)\exp(Y))$ which is usually $X+Y$ except when $XY-YX\neq 0$
$\pi isD*D-D*\pi isD=\pi isD^2-\pi i(1+sD)D=\pi isD^2-\pi iD-\pi isD^2=-\pi iD$
$\exp(\pi isD)\exp(D)=\exp(\pi i(s-\frac{1}{2})D)$
$$\left(2sD-\exp\left(\pi i\left(s-\frac{1}{2}\right)D\right)+1\right)g=0$$
There are infinitely many solutions to this but the easiest is $g=0$.
-
$\begingroup$ I’ll have to read up the links you gave as I’m a high school student and I got lost from “Operator Analysis”. What I do know though is that g=k satisfies the equation where k is any real number. Are u saying there are still infinite solutions other than g=k $\endgroup$ Mar 21, 2020 at 22:38
-
$\begingroup$ Yep. Although why is a bit complicated. It partially has to do with how f(x+1)=f(x) has infinitely many solutions. $\endgroup$ Mar 21, 2020 at 23:20
-
$\begingroup$ f(x) for f(x+1)=f(x) is a periodic function of a periodicity( don’t know if that’s the word) of 1 right? So something like f(x) = sin(2pix) or f(x)=k. How does this relate to my question? If it’s still too complex then it’s okay. I’ve found out what I wanted to. But I’d still like to get it though. $\endgroup$ Mar 21, 2020 at 23:31