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Ordinary and Singular Points

Ordinary differential equations (ODEs) come in many flavours. We will consider scalar ODEs governing a function y(x) of a complex variable x. A general such ODE looks like

y^{(n)}=F(y^{(n-1)}, ..., y', y, x).

We will concentrate on functions F that are rational in y^{(n-1)}, ..., y', y. Moreover, we will assume that F is analytic in x except at a finite number of isolated points x_i, i=0, ..., s-1 where s is a nonnegative integer.

Definition: A point r where F(y^{(n-1)}, ..., y', y, r) is analytic is called an ordinary point of the ODE. The points x_0, x_1, ..., x_{s-1} are called fixed singularities of the ODE.

An example (from Section 0.0 ) is

y'=- y^2/x

Clearly x_0=0 is a fixed singularity.

Excercise: Show that infinity is also a fixed singularity. (Change variables to t=1/x. Show that the resultant ODE for u(t)=y(1/x) also has a fixed singularity at t=0.)

However, any non-zero finite point x is an ordinary point. The main theorem of existence and uniqueness for ODEs shows that every solution defined by an initial value problem given at an ordinary point must be analytic. For the above example, the general solution is

y(x)=1/(log(x)-a).

Define y(1)=1. Clearly the corresponding solution is

y(x)=1/(log(x)+1)

which is clearly analytic at x=1.


Back to Contents Nalini Joshi
Last modified: 12 April 2002 by
N.Joshi