## Ordinary Differential Equation Second-Order

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)

 (3)

Then (1) becomes
 (4)

Case (a): If
 (5) (6)

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
 (15)

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)

 (26)

so

 (27)

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)

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.