Forgotten my maths Simple 1D ODE, spherical coordinates

[JHZR2]

Hi,

I seem to have forgotten some of my math how-to, as I haven't done this in a while. Looking through my notes, Bird, Stewart and Lightfoot, Greenberg, etc. don't really help.

My equation is this, at steady state:

0 = 1/r^2 ∂/∂r (D*r^2 ∂C/∂r) + P

Where P is some production rate.

So the thing that I'm just not remembering is how to deal with the basic function:

∂/∂r (r^2 ∂C/∂r) in terms of how to deal with it to make it solvable.

I was kind of under the impression that I should do a chain rule to make it a real ODE, which gives me:

0 = D*( ∂2C/∂r2 + 2/r ∂C/∂r) + P

Problem is that the 2/r would integrate into log(r). Implementing the BC for the sphere that at r=0, ∂C/∂r=0, I would have an undefined answer since I'd have Log(0).

So something is wrong.

Ive tried searching around, but most places just glaze over the operation to solve this. I don't think it is that hard, but my brain is empty on this at this point - I've forgotten how.

Can anyone advise and/or point me in the right direction?

Thanks very much!

 

[tiny-tim]

Hi JHZR2!

0 = 1/r^2 ∂/∂r (D*r^2 ∂C/∂r) + P

(try using the X² icon just above the Reply box )

Rewrite it as ∂/∂r (D*r² ∂C/∂r) = -Pr², and integrate.