I am currently trying to build a flow chart to visualize all tests there are to tell whether an ordinary differential equation is solvable and how to solve it. This is for tutoring purposes.
The inspiration for this project comes from another flowchart summarizing all tests to tell whether an infinite series converges. http://www.math.hawaii.edu/~ralph/Classes/242/SeriesConvTests.pdf
I would like to make a document similar to this, but instead for ways to solve an ordinary differential equation (or determine that it is not solvable).
Here is what I have so far:
In order to build this, I have written down every method that I know of to solve ODE's and have indicated the situation in which it can be used and the type of solution it gives, in a table.
Here is the link to the chart: https://docs.google.com/document/d/1RYDoOI5Y3eQnEr9WV8b9tlwY4yFBqOIWE3ZIVSp3zCQ/edit?usp=sharing
The trouble is that I am not sure when each of these methods can be used, and which one is preferable if there are multiple approaches that could be used. Additionally, if there are any other methods that you know of that I could add, or any resources that might be useful on this project, then feel free to mention it. It would be helpful if there was a list somewhere online of all known methods of solving an ODE, and especially if it was a more exhaustive list than that on Wikipedia (https://en.wikipedia.org/wiki/Ordinary_differential_equation#Summary_of_exact_solutions).
I wanted to extend the chart's section "Is Solvable" to include more specific tests. Are there any tests I should include besides Picard's theorem?
Lastly, "Matrix methods" is very general. I was wondering if there is a list somewhere online of matrix-based methods that can be used, and when they work.