Consider a solid ball of radius \(R\), with a constant thermal diffusivity \(\kappa\). Initially, the ball has a uniform temperature \(u_{o}\). If this ball is dropped into an ice-water bath of temperature zero, show that the temperature distribution \(u(r, t)\) in the ball can be described by the following partial differential equation when only radial heat conduction is considered: \(\frac{\partial u}{\partial t}=\kappa\left[\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial u}{\partial r}ight)ight]\), with the initial condition \(u(r, t)=u_{o}\), \(00\), and \(u(r, t)=0, r=R, t>0\).

Using the transformation \(\xi=u r\), show that the partial differential equation becomes \(\frac{\partial \xi}{\partial t}=\kappa \frac{\partial^{2} \xi}{\partial r^{2}}\), subject to an initial condition \(\xi(r, t)=u_{o} r, 00\), and \(\xi(r, t)=0, r=R, t>0\). Solve this transformed problem for \(\xi(r, t)\). What is the corresponding solution for \(u(r, t)\) ?