A solid sphere of radius a is immersed in a vat of fluid at a temperature T_0. Heat is conducted into the sphere according to dT/dt = D(d^2T/dr^2) (d-> partial derivative btw) If the temperature at the boundary is fixed at T_0 and the initial temperature of the sphere is T_1, find the temperature within the sphere as a function of time. My reasoning Ok. Here's my reasoning. Use a solution of the form T=X(t)R(r), and plug into the above equation to get R''/R=X/(DX')=-k^2. I get X(t) = C*[exp(-t/(D*k^2))]. (k^2>0 for convergence) Then I have R'' +k^2*R = 0 so R= Acos(kr) + Bsin(kr), since k^2>0. The Problem Assuming that the above steps are right, we could set the boundary conditions and have Acos(ka)+Bsin(ka)=T_0. That only eliminates one variable, either A or B. I don't know where to go from there however. Should this be like a fourier series or something?