316 Chapter 5 Higher Dimensions and Other Coordinates
5.By using Exercise 4 and Rolle’s theorem, and knowing thatJ 0 (x)=0 for an
infinite number of values ofx, show thatJ 1 (x)=0 has an infinite number
of solutions.
6.Using the infinite series representations for the Bessel functions, verify the
formulas
d
dx(
x−μJμ(x))
=−x−μJμ+ 1 (x),d
dx(
xμJμ(x))
=xμJμ− 1 (x).7.Use the second formula in Exercise 6 to derive the integral formula
∫
xμJμ(x)xdx=xμ+^1 Jμ+ 1 (x).8.Use the method of Frobenius to obtain a solution of the modified Bessel
equation (7). Show that the coefficients of the power series are just the
same as those for the Bessel function of the first kind, except for signs.
9.Use the modified Bessel function to solve this problem for the temperature
in a circular plate when the surface is exposed to convection:
1
rd
dr(
rdu
dr)
−γ^2 (u−T)= 0 , 0 <r<a,u(a)=T 1.10.Using the result of Exercise 4, solve the eigenvalue problem
1
rd
dr(
rddrφ)
+λ^2 φ= 0 , 0 <r<a,dφ
dr(a)=^0 ,φ(^0 )bounded.5.6 Temperature in a Cylinder
In Section 5.4, we observed that both the heat and wave equations have a great
deal in common, especially the equilibrium solution and the eigenvalue prob-
lem. To reinforce that observation, we will solve a heat problem and a wave
problem with analogous conditions so that their similarities may be seen.
These examples illustrate another important point: Problems that would be
two-dimensional in one coordinate system (rectangular) may become one-
dimensional in another system (polar). In order to obtain this simplification,
we will assume that the unknown function,v(r,θ,t), is actually independent