Page 59 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 49
3. The differential equation

2

(1 − x )u + u = 0
has singular points at x =±1, but is analytic near x = 0. Find a series solution that is
valid near x = 0 and discuss the radius of convergence.

4.1.4 Regular Singular Points and the Method of Frobenius
If x 0 is a singular point in (4.2) there may not be a power series solution of the form of Eq. (4.3).
In such a case we proceed by assuming a solution of the form

∞
n+r
u(x) = c n (x − x 0 ) (4.18)
n=0
in which c 0 = 0and r is any constant, not necessarily an integer. This is called the method of
Frobenius and the series is called a Frobenius series. The Frobenius series need not be a power
series because r may be a fraction or even negative. Differentiating once

∞

n+r−1
u = (n + r)c n (x − x 0 ) (4.19)
n=0
and differentiating again
∞

(n + r − 1)(n + r)c n (x − x 0 ) n+r−2 (4.20)
u =
n=0
These are then substituted into the differential equation, shifting is done where required so
n
that each term contains x raised to the power n, and the coefficients of x are each set equal
to zero. The coefficient associated with the lowest power of x will be a quadratic equation that
can be solved for the index r. It is called an indicial equation. There will therefore be two roots
of this equation corresponding to two series solutions. The values of c n are determined as above
by a recurrence equation for each of the roots. Three possible cases are important: (a) the roots
are distinct and do not differ by an integer, (b) the roots differ by an integer, and (c) the roots
are coincident, i.e., repeated. We illustrate the method by a series of examples.
Example 4.3 (distinct roots). Solve the equation

x u + x(1/2 + 2x)u + (x − 1/2)u = 0 (4.21)

2
The coefficient of the u term is

(1/2 + 2x)
p(x) = (4.22)
x