An ODE
|
(1) |
has singularities for finite under the following conditions: (a) If either or diverges as , but and remain finite as , then is called a regular or nonessential
singular point. (b) If diverges faster than so that
as , or
diverges faster than so that
as , then is called an
irregular or essential singularity.
Singularities of equation (1) at infinity are investigated by making the substitution
, so
, giving
|
(2) |
Then (1) becomes
|
(4) |
Case (a): If
remain finite at
(), then the point is ordinary. Case (b): If either diverges no more
rapidly than or diverges no more rapidly than , then the point is a regular singular point.
Case (c): Otherwise, the point is an irregular singular point.
Morse and Feshbach (1953, pp. 667-674) give the canonical forms and solutions for second-order ODEs classified by
types of singular points.
For special classes of second-order linear ordinary differential equations, variable Coefficients can
be transformed into constant Coefficients. Given a second-order linear ODE with variable
Coefficients
|
(7) |
Define a function ,
|
(8) |
|
(9) |
|
(10) |
|
(11) |
This will have constant Coefficients if and are not functions of . But we are free to set
to an arbitrary Positive constant for by defining as
|
(12) |
Then
|
(13) |
|
(14) |
and
Equation (11) therefore becomes
|
(16) |
which has constant Coefficients provided that
|
(17) |
Eliminating constants, this gives
|
(18) |
So for an ordinary differential equation in which is a constant, the solution is given by solving the second-order
linear ODE with constant Coefficients
|
(19) |
for , where is defined as above.
A linear second-order homogeneous differential equation of the general form
|
(20) |
can be transformed into standard form
|
(21) |
with the first-order term eliminated using the substitution
|
(22) |
Then
|
(23) |
|
(24) |
|
(25) |
so
Therefore,
|
(28) |
where
|
(29) |
If , then the differential equation becomes
|
(30) |
which can be solved by multiplying by
|
(31) |
to obtain
|
(32) |
|
(33) |
|
(34) |
If one solution () to a second-order ODE is known, the other () may be found using the Reduction of
Order method. From the Abel's Identity
|
(35) |
where
|
(36) |
|
(37) |
|
(38) |
|
(39) |
But
|
(40) |
Combining (39) and (40) yields
|
(41) |
|
(42) |
Disregarding , since it is simply a multiplicative constant, and the constants and , which will contribute a
solution which is not linearly independent of ,
|
(43) |
If , this simplifies to
|
(44) |
For a nonhomogeneous second-order ODE in which the term does not appear in the function ,
|
(45) |
let , then
|
(46) |
So the first-order ODE
|
(47) |
if linear, can be solved for as a linear first-order ODE. Once the solution is known,
|
(48) |
|
(49) |
On the other hand, if is missing from ,
|
(50) |
let , then , and the equation reduces to
|
(51) |
which, if linear, can be solved for as a linear first-order ODE. Once the solution is known,
|
(52) |
See also Abel's Identity, Adjoint Operator
References
Arfken, G. ``A Second Solution.'' §8.6 in Mathematical Methods for Physicists, 3rd ed.
Orlando, FL: Academic Press, pp. 467-480, 1985.
Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 4th ed.
New York: Wiley, 1986.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 667-674, 1953.
© 1996-9 Eric W. Weisstein
1999-05-26