Computer Algebra Recipes

136 CHAPTER 3. LINEAR ODE MODELS
Q0 = 1
1
ln(x +1)¡
2 2 ln(1 ¡ x); Q1 = 1
1
x ln(x +1)¡ x ln(1 ¡ x) ¡ 1;
2 2
Q2 = ¡ 1
ln(x +1)+1 ln(1 ¡ x)+3
4 4 4 x2 ln(x +1)¡ 3
4 x2 3 x
ln(1 ¡ x) ¡
2
The Qm(x) formequal to zero or a positive integer diverge to in¯nity at x = ¡1
and +1 and must therefore be rejected as being unphysical.
PROBLEMS:
Problem 3-15: A Sturm{Liouville equation
Show that the following ODE is of the Sturm{Liouville form:
y 00 (x) ¡ 2 xy 0 (x)+2ny(x) =0:
Determine the general solution of this ODE and plot the included special functions
over a suitable range of the independent variable x.
Problem 3-16: Recurrence Formula
Bessel functions of di®erent orders can be related through recurrence relations.
Use Maple to prove the following Bessel function recurrence relations.
² Jm¡1(x)+Jm+1(x) =(2m=x) Jm(x);
² 4 d2Jm(x) dx 2 = Jm+2(x) ¡ 2 Jm(x)+Jm¡2(x).
Problem 3-17: Bessel function solutions
Find the general solution of each of the following ODEs in terms of Bessel
functions and identify the order. Identify any other new functions that occur.
(a) y00 (x)+y(x)= p x =0;
(b) y00 (x)+xy(x) =0, Hint: Useconvert( ,Bessel);
¡ x 8
5 y(x) =0;
(c) d2
dx 2
μ
x 16
5 d2 y
dx 2
Problem 3-18: Orthogonality
An important general property that all solutions yn(x) of the Sturm{Liouville
equation corresponding to a given ¸n possess is orthogonality. Provided that
y(x) ory 0 (x) orp(x) vanishes at the endpoints a and b of the range (referred
to as Sturm{Liouville boundary conditions), then
Z b
a
w(x) ym(x) yn(x) dx =0; for m 6= n; (3.3)
where w(x) isreferredtoastheweight function. Con¯rm the orthogonality
property for Legendre functions of the second kind of orders 2 and 3 over the
range x = ¡1 to +1. Which of the possible Sturm{Liouville boundary conditions
is satis¯ed?