@Simon45 As you yourself said, "the advantage of the Milne method over [RK4] is a higher speed." My guess is that backtracking is done in the hope that future iterations can be avoided. If you set eps lower, you'll see an increase in the number of iterations as measured by the number of actual calls to the derivative evaluator (counted by the trick that I gave). This suggests to me that the method tries to achieve a target accuracy and can forego some iterations if they're not needed. Note that for most methods that use a fixed stepsize (your h), such as RK4, the number of iterations is determined by that stepsize rather than by a tolerance (your eps).

@mimi990 Try separating the symbolic/scalar and Matrix terms and generating Matlab code for them separately.

Your code works perfectly for me in Maple 2019. I get the following plot, which shows that the approximation error is below the limit of visual detection:

What happens when you run it in your Maple?

@mmcdara By "aliasing", do you mean Gibbs phenomenon?

For the problem at hand, I think that it's clear (by default) that the intention is to consider the function as periodic with period 2*Pi in the simplest way: simply with jump discontinuities at 2*Pi*n for all integer n.

Is MVA millivolt-amps? megavolt-amps? What is the r in MVAr?

@mmcdara You need to replace all occurences of point with geometry:-point.

While the geometry package is quite old, and it has significant differences from more-modern packages, I don't see this as one of those differences.

@tizozadoxo I gave a clear method for doing this in Maple. What don't you understand about what I wrote? I'd be happy to give more details.

@Kitonum Oh, I see them now. I overlooked them as dirt on my screen. I guess that they're not supposed to be there.

I tried to answer your question, but I'm confused by the plethora of d-like symbols in your expressions. Do they all represent derivatives/differentials? In particular, what's the lowercase delta?

@jmalik7 Don't confuse FAIL with false. FAIL means that is was neither able to rigorously prove the universally quantified statement (the quantification being with respect to the assumptions on symbolic variables) nor was it able to find a counterexample. Since finding counterexamples (when they exist) is generally easier than proving a universal, FAIL results are more likely to be in reality true rather than false. 

Acer said, "you may receive a result of FAIL from is, which doesn't help." Yes, it doesn't help if you require rigor. But if just knowing what's likely is sufficient, I'd accept FAIL as true. Likewise, a FAIL from coulditbe (the existential quantifier version of is) is likely to be false in reality.

@Ronan The number of solutions is given by a well-known function of the modulus called the totient, Euler's totient, or Euler's phi. You should read it about it in Wikipedia. It may be the most important function in number theory. It is a simple function of the prime factorization of the modulus. Since 10000 = 2^4*5^4, its totient is (2-1)*2^(4-1)*(5-1)*5^(4-1) = 4000. Via Maple, it's NumberTheory:-Totient(10000).

This only works because 2391 and 10000 are relatively prime. If the coefficient and the modulus aren't relatively prime, the situation is more complicated.

@dharr IIRC, more than 15 years ago, the sole difference was that the maximum setable value of Digits was 100. That difference has long since been dropped.

@dharr AFAIK, there are no differences (other than the price) between the student edition Maple and the much-more-expensive editions.

I don't understand why VV's msolve solution has received three upvotes while mine has received none. My solution is well over 1000 times faster and just as easy to use and to understand.

@vv I guess that I should've said

• I don't think that it actually matters whether we consider x to be an independent variable or a constant.

If x is considered a dependent variable, as you suggest, then, yes, I agree that that would make a difference.