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Motivation: This was a printing error in my MCQ test, correct question contained the first order derivative. But it's destroying my mental peace.

My attempt

I don't even know where to start. This looks so utterly simple yet it isn't. Only thing I could do is

Let $y=f(x)$. So $$\frac{d^2y}{dx^2}=\frac{d\bigg(\frac{dy}{dx}\bigg)}{dx} \implies d^2 y \ dx =d\bigg(\frac{dy}{dx}\bigg) dx^2$$

How to proceed anywhere from here?

Gwen
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    You cannot proceed on your current path, as I explained why here – Ninad Munshi May 18 '24 at 14:11
  • @NinadMunshi thanks for that. If I correct my path now, still will I be able to get a closed form solution? – Gwen May 18 '24 at 14:37
  • If $f\propto\exp\int gdx$ so $f'=gf$ your integral becomes $\int(g^{\prime\prime}+g^{\prime2})dx$, but the second term is intractable in general. – J.G. May 18 '24 at 15:28
  • apparently MCQ means multiple choice questions – Will Jagy May 18 '24 at 16:16
  • @WillJagy yessir. But it wasn't the original question, it was a typo. Original answer had the usage of $\ln f(x)$ to solve a logarithmic sum – Gwen May 18 '24 at 18:12
  • @NoChance would you like to show your manipulation as an answer? I can't bring it to my desired form. Whatever I do to $\frac{f'(x)}{f(x)}=\frac{f'(x)}{f(x)}$ it stays the same. Double differential just cancels out – Gwen May 19 '24 at 03:12
  • OK, but the result on the R.H.S will not be an integral that can be evaluated analytically. I mean we will end-up with difficult form=difficult form, is this something you are interested in? – NoChance May 19 '24 at 09:52
  • Hi @Gwen, what if we did this - $\int \frac{f''(x)}{f(x)} dx=\int \frac{f''(x)}{f'(x)}\cdot\frac{f'(x)}{f(x)},dx$ and maybe try out IBP or something.. – Amrut Ayan Dec 20 '24 at 16:08

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