Gamma Function Q from Mary Boas 2nd ed (ch11)

[weak_phys]

Homework Statement

So this Q has been nagging me for a few days & I noticed that some poor citizen from 2006 has asked about it previously.
https://www.physicsforums.com/threads/gamma-function-application.139116/
But in spite of the advice from OlderDan, i'm not seeing how to manipulate to find the Gamma integral form - any further hints appreciated (but its not actually homework so no panic, I was just working my way through this chapter)

Relevant Equations

16. A particle starting from rest at x=1 moves along the x axis toward the origin. Its potential energy is $$V= \frac{1}{2} m lnx$$, Write the Lagrange equation and integrate it to find the time required for the particle to reach the origin. Answer is given $$\Gamma(\frac{1}{2})$$

So using $$L=\frac{mv^2}{2} - \frac{1}{2} m lnx$$ and throwing it into the Euler-L equation I agree with kcrick & OlderDan that we can manipulate this to either $$\frac{d}{dt} m\dot{x} = -\frac{m}{2x}$$ or $$2vdv = -\frac{dx}{x}$$ but I'm not having any epiphanies on how to turn the above into something like $$ \int_{0}^{\infty}\frac{1}{\sqrt{x}}e^{-x}dx$$ or $$ \int_{0}^{1} [ln2]^{-\frac{1}{2}}dx$$
Again, I'm a newbie, any help appreciated or if I've posted in the wrong place, please forgive. My aim here is to eventually be the kind of math teacher that Mary Boas likens in her preface: "What do you say when students ask about the practical applications of some mathematical topic?" The experienced professor said "I tell them!"
In uni we studied the use of Gamma functions as part of Bessel function solutions for wavey pde's but...
I like this question because it's 'simple' and because of the shape of the potential and it bothers me that I'm getting nowhere :(

 

[Antarres]

Integrating your second equation, you get:
$$\int 2vdv = \int (-\frac{dx}{x}) + C$$
or
$$v^2 = -\ln(x) + C$$.
Set $v=0$ at $x=1$, you get $C = 0$. Then you use $v = \frac{dx}{dt}$ and you integrate again. You will get the form of Gamma function that was mentioned in that post.

 

[weak_phys]

Ah, thank you. I think I was having a mental block by insisting I have a definite integral from $$\int_{1}^{0} - \frac{1}{x} dx$$ in the first step which stopped any progress. That's excellent, thank you again, I really appreciate your help!

 

[Antarres]

You're welcome