Consider a solid spherical metal ball of radius a and let u = u(r, t) be the spherically symmetric temperature distribution inside the ball. %3D a. Assuming the separable solution u(r, t) = R(r)G (t), derive the differential equations governing R and T. You may assume that the separation constant is non- positive. b. By letting R(r) = p(r)/r, derive the differential equation governing p for r>0, and hence, find the general solutions for R(r). c. Supposing that the surface of the metal ball is held at 0°C, derive the general solution for u(r, t). d. Given that the initial temperature inside the metal ball is 200 u(r, 0): sin COS 2a 2a Determine the temperature distribution u (r, t) at any time t > 0.

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.