334 Chapter 5 Higher Dimensions and Other Coordinates
EXERCISES
1.Find the general solution of the differential equation
(
xnφ′)′
+λ^2 xnφ= 0 ,wheren= 0 , 1 , 2 ,....
2.Find the solution of the equation in Exercise 1 that is bounded atx=0.
3.Find the solutions of Eq. (9), including the caseλ^2 =0, and prove that
Eq. (12) is a solution of Eqs. (2)–(4).
4.Show that any function of the formu(ρ,t)=^1 ρ(
φ(ρ+ct)+ψ(ρ−ct))
is a solution of Eq. (13) ifφandψhave at least two derivatives.
5.Find functionsφandψsuch thatu(ρ,t)asgiveninExercise4satisfies
Eqs. (14)–(16).
6.Give the formula for thea’s andb’s in Eq. (12).
7.What is the orthogonality relation for the eigenfunctions of Eqs. (18)–
(20)? Use it to find thea’s andb’s in Eq. (22).
8.Sketch the first few eigenfunctions of Eqs. (18)–(20).
9.Find the functionv(x)that is the solution of Eqs. (26) and (27).
10.Use the technique of Example C to change the following problem into a
potential problem:
∂^2 u
∂x^2+∂
(^2) u
∂y^2
=−f(x), 0 <x<a, 0 <y<b,
u=0 on all boundaries.
11.In Exercise 10, will the same technique work iff(x)is replaced byf(x,y)?
12.Verify that Eqs. (31) and (32) form a regular Sturm–Liouville problem.
Show the eigenfunctions’ orthogonality by using the orthogonality of the
Bessel functions.
13.Find a formula for theanof Eq. (35).
14.Verify that Eq. (34) is a solution of Eqs. (28)–(30).