# Diagram of Methods to Solve Differential Equations

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.

Thanks

• Your diagram has the significant weakness that it does not show the answers to the questions, which should be on the lines that come out of the question. Also confusing, yes-or-no questions have more than two lines coming out of them. – Rory Daulton Jul 19 at 12:21
• For non-numerical methods you might deriving a chart from a book like archive.org/details/elementarytreati032501mbp/page/n13 (which isn't quite as "elementary" as its title might suggest) – alephzero Jul 19 at 22:32
• "Matrix methods" is a very general term. You could draw up a whole flow chart for each of several different approaches (finite element, finite difference, symplectic integrators, state-space methods, transfer functions, etc, etc...) – alephzero Jul 19 at 22:37

## 1 Answer

I would just like to mention that other similar flowcharts have been developed, of varying degrees of generality, which you might consult. Here is one (by Adam Monahan). And another (by Jeremy Higgins): And another (by Enrique Areyan): • The second chart only works for the examples covered in the guy's course, apparently. For example, how do you "solve the auxiliary equation" for a second order PDF when $p$ and $q$ are both functions of $x$ and $y$, not constants :) – alephzero Jul 19 at 22:39
• @alephzero: I have to admit I did not study the charts closely, just noted that there are many out there for comparison. – Joseph O'Rourke Jul 20 at 0:16