introductory examples
regular singular points
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
\( y^{(n)}=F(y^{(n-1)}, ..., y', y, x)\qquad (0.3)\)
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.
Equation (0.2) (from Section 0.0 ) is\( y'=- y^2/x\qquad\qquad (0.2) \)
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 a regular initial value problem given at an ordinary point must be analytic. For Equation (0.2), 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 analytic at the initial point \( x=1\), has a logarithmic singularity at \(x=0\), and has a pole when \(\log(x)=-1\), i.e., \(x=e^{-1}\).If we change the initial condition to \(y(1)=-1\), the corresponding solution becomes
\( y(x)=1/(\log(x)-1)\)
This solution is analytic at the initial point \(x=1\). It still has a logarithmic singularity at \(x=0\), but its pole now occurs where \(\log(x)=+1\), i.e., \(x=e^{1}\). It is clear that the pole has moved to a different location.
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Nalini Joshi
Last modified: 12 October 2011 by N.Joshi