Solving a System of Differential Equations

edit

That function desolve_system_rk4 is very badly documented. I would suggest filing a bug report on the documentation.

That said, I think the answer to your question is that you have entered the DE in the wrong way. The list of expressions you pass to the solver need to be the right hand sides of equations you get after solving for $X'(t)$ and $S'(t)$ respectively. So instead of $\frac{-X'(t)}{Y}$ for the second expression, you want $$\frac{-0.01XS}{2+S}$$ . Then you don't need to define $X(t)$ and $S(t)$ as symbolic functions (which is I think where the error lies in your screenshot), they can remain just "variables". Here's an example:

sage: (X,S,t) = var('X,S,t')
sage: des = [0.01*X*S/(2+S), -0.01*X*S/(2+S)]
sage: desolve_system_rk4(des, [X,S], ics=[0,0.1,10], ivar=t, end_points=20)
[[0, 0.100000000000000, 10], [0.1, 0.100083368007, 9.99991663199], 
 [0.2, 0.100166805401, 9.9998331946], [0.3, 0.100250312238, 9.99974968776], 
 [0.4, 0.100333888577, 9.99966611142], [0.5, 0.100417534474, 9.99958246553], 
...

I don't know how to do this. Could you try re-defining f in the first place, without using mu as a variable but instead using the expression you've given?

Also, your question might be easier for someone else to answer if you copy/paste the function definitions (using the code format button to format it correctly here). Then people who want to help can just copy/paste to get their own versions to test with.