Second-Order Ordinary Differential Equation

An ordinary differential equation of the form

Such an equation 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

Then (3) becomes

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 ordinary differential equations classified by types of singular points.

For special classes of linear second-order ordinary differential equations, variable coefficients can be transformed into constant coefficients. Given a second-order linear ODE with variable coefficients

Define a function ,

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

Then

and

Equation (◇) therefore becomes

which has constant coefficients provided that

Eliminating constants, this gives

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

for , where is defined as above.

A linear second-order homogeneous differential equation of the general form

can be transformed into standard form

with the first-order term eliminated using the substitution

Then

so

Therefore,

where

If , then the differential equation becomes

which can be solved by multiplying by

to obtain

For a nonhomogeneous second-order ordinary differential equation in which the term does not appear in the function ,

let , then

So the first-order ODE

if linear, can be solved for as a linear first-order ODE. Once the solution is known,

On the other hand, if is missing from ,

let , then , and the equation reduces to

which, if linear, can be solved for as a linear first-order ODE. Once the solution is known,

Nonhomogeneous ordinary differential equations can be solved if the general solution to the homogenous version is known, in which case variation of parameters can be used to find the particular solution. In particular, the particular solution to a nonhomogeneous second-order ordinary differential equation

can be found using variation of parameters to be given by the equation

where and are the homogeneous solutions to the unforced equation

and is the Wronskian of these two functions.