201 sin np P =12. n=1.2... EXERCISES Show that the eigenvalue problem (4.68 4.69) has no negative eigenvalues. 2 la the cooling of the sphere problem take Rr inches, To 37 degrees Celsius, and k-5.58 inches-squared per hour. Using (4.70), plot the tem perature at the center of the sphere as a function of time; take t=0 to ercise to show that there are no negati P) = solutions to the time equation for g Thus we have determined infinitely sin np =1,2..... be boundary conditions. To satisfy the ear combination sin no This problem deals with the cooling of a sphere with a radiation boundary condition. A spherical ball of radius R has a diffusivity k and initial tem- perature u f(p), 0P5 R, which depends only on the distance p from the center. Heat radiates from the surface of the sphere via the law -1 hour. up(R,t) -hu(R,t), where h is a positive constant and Rh <1. Find a formula for the temper- ature u(p, t) in the sphere. Hint: Find the eigenvalues A,, as the positive roots of the equation tan RX= RX 1-Rh Next, take R = inches, k 5.58 inches-squared per hour, f(p) = 37 degrees Celsius, and h = 0.1 per inch, and calculate a four term approxi- mate solution to the temperature function. sin np hp