(Solved): Problem 4. Recall that the formula for the Laplacian in spherical coordinates is \[ \nabla^{2} w=\ ...

Problem 4. Recall that the formula for the Laplacian in spherical coordinates is \[ \nabla^{2} w=\frac{1}{\rho^{2}} \frac{\partial}{\partial \rho}\left(\rho^{2} \frac{\partial w}{\partial \rho}\right)+\frac{1}{\rho^{2} \sin \theta} \frac{\partial}{\partial \theta}\left((\sin \theta) \frac{\partial w}{\partial \theta}\right)+\frac{1}{\rho^{2}(\sin \theta)^{2}} \frac{\partial^{2} w}{\partial \phi^{2}} . \] Here \( w(\theta, \phi, \rho) \) is a function of \( \rho \in[0,+\infty), \theta \in[0,2 \pi) \) and \( \phi \in[0, \pi] \). Consider a spherical shell between the two concentric spheres; the smaller sphere has the radius 1 and the larger sphere has the radius 2. Assume that the smaller sphere is kept at the temperature \( T_{1} \) and the larger sphere is kept at the temperature \( T_{2} \). Determine the equilibrium temperature distribution of this spherical shell.