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. 667674) give the canonical forms and solutions for secondorder ODEs classified by
types of singular points.
For special classes of secondorder linear ordinary differential equations, variable Coefficients can
be transformed into constant Coefficients. Given a secondorder 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 secondorder
linear ODE with constant Coefficients

(19) 
for , where is defined as above.
A linear secondorder homogeneous differential equation of the general form

(20) 
can be transformed into standard form

(21) 
with the firstorder 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 secondorder 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 secondorder ODE in which the term does not appear in the function ,

(45) 
let , then

(46) 
So the firstorder ODE

(47) 
if linear, can be solved for as a linear firstorder 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 firstorder 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. 467480, 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: McGrawHill, pp. 667674, 1953.
© 19969 Eric W. Weisstein
19990526