State the governing equation and the dimensionless parameters needed to characterize the problem.

Verify that the temperature profile is given by

\[\theta=\theta_{\max } \frac{J_{0}(\zeta \sqrt{\beta})-J_{0}(\sqrt{\beta})}{1-J_{0}(\sqrt{\beta})}\]

Show that it satisfies the differential equation and the boundary conditions. Also derive an expression for the maximum temperature, \(\theta_{\max }\). Find the limit for no explosion. The answer is \(\sqrt{\beta}<2.4048\).