singularity and degeneracy of an ODE

Question

I have trouble distinguishing the difference of singularity and degeneracy in the context of ODE theory. Could anyone give me a couple of examples in illustrating the difference of singular point and the point of degeneracy of an ODE?

Answer 1

If you have $P(x)y''+Q(x)y'+R(x)y=0$ then it has a singularity when $\frac{Q(x)}{P(x)}$ and $\frac{R(x)}{P(x)}$ are not defined

May be there is a simple and single definition for Degeneracy but i am not aware or able to word what i know properly hopefully the following helps.

Degeneracy is a more complicated depending on equation i would say there are multiple interpretations but generally the most basic one when you have a repeating root for a characteristic equation of homogeneous part of ODE you have degeneracy since the solutions are not linearly dependent so to avoid this and we look for linearly independent solution.

for example if the root $r$ is repeated $k$ times then we generally look for solutions of the form $A_1e^{rx}+A_2 xe^{rx} +A_3 x^2 e^{rx}+....+A_n x^{n-1} e^{rx}$ as these may be linearly independent again if some of these are dependent you need to keep on increasing powers until you get a set of $k$ linearly independent solutions.

Even if we don't have repeated roots our homogenous solutions might be the same as the non-homogenous part which causes us problems so again we say we have a degenerate case.

Also when you draw phase planes for this ODE we usually call this root as Degenerate Stable/unstable node.

If we have a system of equations if the Jacobean is zero then we say its a degenerate system.