I want to apply the 10th-order Runge-Kutta method, but I am having trouble finding the coefficients. I read Ernst Hairer's article, he used the stage s=17 and k>=10. I tried solving the equations in Python with the scipy library (fsolve, root, newton_krylovl) but didn't get the same results.
Does anyone have knowledge of how to solve the equations or have they seen code that solves it?
Julia's DifferentialEquations.jl package (linked here via the subpackage OrdinaryDiffEq.jl which contains the source code of many of the methods) includes an implementation of a 10th order explicit RK method based on Feagin, 2012 (pdf link), as well as many other performant high-order RK methods.
You seem to be misunderstanding. Since you're referring to an explicit Runge-Kutta method, applying the method does not require that you solve any equations.
In Ernst Hairer's article, he is deriving such a method, so he is solving the order conditions. This has essentially nothing to do with implementing the method to solve differential equations.
As others pointed out, Hairer's article already contains the coefficients, there is no need to solve for them. They are in Table 3, which is at pages 12-13 of the PDF (p.57-58 if you follow the journal's pagination).
I wanted to point out that if you are interested in how to solve for these coefficients in detail, you can look at Peter Stone's webpage for this scheme, where there is a very detailed Maple worksheet, and also a PDF summarizing a few things about it. [As a note, it seems like a few coefficients in that PDF differ slightly from Hairer's paper like $c_2$, $a_{2,1}$, $a_{3,1}$, $a_{3,2}$, etc., due to some differences in the conditions used in the solving process, but the paper's coefficients are in the Maple worksheet.]
