# Didactic examples in linear ordinary differential equations

I believe it is very common among beginners in differential equations to feel like this field is absolutely chaotic and that there is little to no theory and only techniques which may be applicable only in some rare cases. Recently however I have come to believe that for the case of linear ordinary differential equations one can do much better than just a a bag of tools.

What are the most simple instructive examples in the theory of linear ordinary differential equations?

I use "instructive" in the sense that they contain an instance of a general qualitative phenomenon which occurs in many different equations.

For instance here is a very partial list I made for myself. All will be of the form $Ly=0$ and so i will only specify $L$.

1. The fractional monomial $L=x \partial + c$ - regular singularities at 0 and infinity. If $c \ne 0,1$ solutions have branch points. If $c \notin \mathbb{Q}$ solution is transcendental.
2. The exponentiated monomial $L = x^{n}\partial + c, n \in \mathbb{Z} \text{\\}\{1\}$ - Irregular singularity at zero/infinity.
3. The Legendre equation $L=(1-x^2)\partial^2 -2x\partial+c$ - Regualr singularities at $-1,1, \infty$. One solution will be entire, the other will have singularities at $-1,+1,\infty$, non trivial monodromy representation.
4. The Airy equation $L=\partial^2 - x$ - Irregular singularity at $\infty$, stokes phenomenon, trivial monodromy.
5. The Bessel equation $L=\partial^2+x^{-1}\partial +(1-n^2x^{-2})$ - regular singularity at $0$ and irregular singularity at $\infty$. Stokes phenomenon and non-trivial monodromy.

I'm sure the above partial list is very naive and elementary i think a a list like this written by a specialist could be of a lot of help to anyone willing to put some effort in studying said examples.

• What would constitute a full answer to your question? – Χpẘ Jun 1 '17 at 17:02
• @Χpẘ I was thinking of smoething more along the long of the following questions (which although are about a different topic were of a lot of help to me) mathoverflow.net/questions/34110/algebraic-geometry-examples mathoverflow.net/questions/63033/… math.stackexchange.com/questions/666113/… – Saal Hardali Jun 1 '17 at 23:33
• I guess that your list will very efficiently reinforce that feeling of chaos ! – Yves Daoust Jun 4 '17 at 16:47
• Instead of asking for didactic examples, I would rather look for a general framework. Such a framework is Lie Group Theory, which has been designed originally for the purpose of classifying and solving differential equations in a systematical manner. Unfortunately I can't provide Mathematics Stack Exchange references covering your area of interest, but perhaps the following (chain) reference(s) may be helpful : How to derive these Lie Series formulas . – Han de Bruijn Jun 5 '17 at 16:07
• And this book contains the general framework I've been talking about: Sophus Lie, Vorlesungen uber Differentialgleichungen mit bekannten Infinitesimalen Transformationen, bearbeitet und herausgegeben von Dr. Georg Wilhelm Scheffers, Leipzig (1891). Availability: Amazon . – Han de Bruijn Jun 5 '17 at 16:10