Bessel's Differential equation

[Fisherman87]

Homework Statement

A solid ball at 30 degrees C with radius a=1 is placed in a refrigerator that maintains a constant temperature of 0 degrees C. Take c (speed)=1 and determine the temperature u(r,theta,phi,t) inside the ball

Homework Equations

partial differential heat equation in spherical coordinates

The Attempt at a Solution

In this case, I started out with setting all the partials of the function u (assuming u is the solution) with respect to theta and phi as 0 since the temperature is uniform and doesn't depend on either. I then set u(r,t)=R(r)T(t), and then plugged back into the partial equation in order to separate variables, and I got T(t)=exp(mu*t). However, R turns into a bessel equation that I can't figure out how to solve, with the coefficient of R double prime equal to r^2, the coefficient of R prime being 2r, and the coefficient of R being mu. Mu is the separation constant. The book is generally unhelpful. Can anyone tell me how to solve that equation?

 

[LCKurtz]

Homework Statement

A solid ball at 30 degrees C with radius a=1 is placed in a refrigerator that maintains a constant temperature of 0 degrees C. Take c (speed)=1 and determine the temperature u(r,theta,phi,t) inside the ball

Homework Equations

partial differential heat equation in spherical coordinates

The Attempt at a Solution

In this case, I started out with setting all the partials of the function u (assuming u is the solution) with respect to theta and phi as 0 since the temperature is uniform and doesn't depend on either. I then set u(r,t)=R(r)T(t), and then plugged back into the partial equation in order to separate variables, and I got T(t)=exp(mu*t). However, R turns into a bessel equation that I can't figure out how to solve, with the coefficient of R double prime equal to r^2, the coefficient of R prime being 2r, and the coefficient of R being mu. Mu is the separation constant. The book is generally unhelpful. Can anyone tell me how to solve that equation?

The Bessel's equation is solved with series methods. You can read about how to do it here:

http://www.ucl.ac.uk/~ucahhwi/MATH7402/handout9.pdf

 

[Fisherman87]

I guess I should've been more descriptive. I have all that information already (even though this is for spherical coordinates and not polar). The equation I got for the last term multiplied by R is mu*r^2. This is why I'm having trouble, since it doesn't really agree completely with the normal bessel equation, and I don't understand how the helmholtz equation is related to this equation, and why I should use it.