soln of (1/r)(d/dr)(r(dT/dr))=0, T(r1)=T1, T(r2)=T2 is....

[f0rked]

Hi,
I'm trying to figure out how to get from the initial differential equation through to the solution, and I'm totally stumped. Can't seem to find anything remotely helpful in any of my maths textbooks, so I'm hoping someone here can shed some light on this;

For instance, I'm not really sure where to start. Do I separate this so there is a function on either side, or expand the brackets, or what? I think I'm meant to set it all = T on the RHS, but again, I'm really not sure what to do after that (or why it's T1-T2 instead of the other way around..). It would be great if someone could show me how to start this/point me in the right direction.

Thanks in advance!

 

[Deleted member 4993]

Re: Not sure where to start..

Hi,
I'm trying to figure out how to get from the initial differential equation through to the solution, and I'm totally stumped. Can't seem to find anything remotely helpful in any of my maths textbooks, so I'm hoping someone here can shed some light on this;

For instance, I'm not really sure where to start. Do I separate this so there is a function on either side, or expand the brackets, or what? I think I'm meant to set it all = T on the RHS, but again, I'm really not sure what to do after that (or why it's T1-T2 instead of the other way around..). It would be great if someone could show me how to start this/point me in the right direction.

Thanks in advance!

Start with solving

d/dr [r dT/dr] = 0

show us what do you get - and exactly where you get stuck.

The solution to this problem is related to the other problem you had posted:

viewtopic.php?f=3&t=28495

 

[f0rked]

Re: Not sure where to start..

Okay, So I tried that and got this:

,
which, if I add in the 1/r, gives me:

I have a feeling I'm going wrong somewhere in the first step though, or (more likely) even in both! Has been a while since I've had to know this, and trying to refresh my memory for one of my courses!

 

[Deleted member 4993]

\(\displaystyle \frac{d}{dr}[r \frac{dT}{dr}] \,= \,0\)

\(\displaystyle r \frac{dT}{dr}\, = \,C_1\)

\(\displaystyle C_1\cdot\,\frac{dr}{r}\, =\, dT\)

\(\displaystyle T - T_1 = \,C_1\cdot ln(\frac{r}{r_1})\)

Now continue....