@Kitonum No, my plot doesn't prove it, nor did I say that it did. The presence of decimal approximations in the original makes it impossible to give a definite answer. The differences in the zeroes of the numerator and denominator (after applying convert(..., rational)) occur after the 10th decimal place. Given that this problem has some real-world context, it's reasonable to assume that those zeroes occur at exactly the same values and any apparent discrepancy is merely due to rounding error.

@mehwish Were you able to sufficiently understand my explanation? And, more importantly, were you able use my code to achieve your desired outcome?

@wolfgan Intuitively, you should get circular motion around the z-axis whenever

D(phi)(0) <> 0 #Initially there's some horizontal motion
theta(0) > 0 #Initially there's some vertical displacement
and
D(theta)(0) = 0 #Initially there's no vertical motion

@Carl Love Taking a closer look, it seems that the discontinuities are indeed removable, and so the RMS is indeed the finite value that I initially reported. Consider this plot:

plot([10^(-5)*denom, numer](simplify(x(t)), t= -P..P);

The presence of floating-point constants makes it difficult to say with certainty that the zeroes of the numerator and denominator occur at exactly the same values of t, but certaintly they're close enough.

@janhardo First, note that the word series is both singular and plural. There is no such word "serie". Power series can be differentiated and integrated termwise. So, for example,

1/(1+x) = Sum((-1)^n*x^n, n= 0..infinity)  #basic geometric series.

Integrate both sides (with respect to x): 

ln(1+x) = Sum((-1)^n*x^(n+1)/(n+1), n= 0..infinity)

Given the types of problems that you're working on, you should have a good three-semester calculus textbook for reference. This Betounes's book seems heavily weighted towards calculus and plotting. I can hardly see how it relates to programming in general. If you had a calculus textbook, you'd see that there's a section named "Differentiation and integration of power series".

And I disagree totally about complex analysis. It's much easier than real analysis. There are little to no weird counterexamples. If a function has a first deriavative, then it necessarily has derivatives of all orders. 

@mehwish P, V, and t only occur inside a procedure (delimited by proc ... end proc). It's very important for you to understand that you don't need to understand how a procedure works in order to use it, just like you don't need to know how the engine of a car works in order to drive it.

P and V are parameters of the procedure. They will be substituted by arguments when the procedure is used. In the given example, P will become ODE, and V will become {Q(x), diff(Q(x), x)}. The t is a local of the procedure. It will become the sequence of V-monomials in P stripped of their coefficients. This is explained further on help page ?coeffs.

@mehwish Yes, I figured out what you meant. My Answer below already covers the case in your example. I was simply thrown off by the word "iteration".

@janhardo You can safely just ignore what I said about complex cases. I just felt a need to say it for the sake of completeness. The complex numbers are always "lurking" in the background when you're dealing with power series and analytic functions.

@mehwish Okay, I'm understanding better now. By Q², for example, you mean Q(x)*Q(x) (2nd power), not Q(Q(x)) (2nd iteration), right? I was thrown off by your initial use of the word "iteration".

Are you the author of the PDF that you attached?

@Rouben Rostamian  While I agree that it's a good idea to declare the index variable local, it doesn't seem to help in this case, as my procedure example shows. You can't declare an indexed variable local; you can only declare its "parent" or "stem" name.

Nor does this help to explain the anomaly, because it's manifested exactly the same way whether the index is global or local.

I don't think that anything is possible along those lines if you don't supply an explicit formula for Q.

@Carl Love There's a good chance that the Answer above is wrong due to the discontinuity issue pointed out by Kitonum. I also noticed the discontinuities and the infinity returned from some numeric integrals, but I ignored these because supericial plotting indicated that the discontinuities were removable. Perhaps that was foolish.

@Mohamed Abohamer What is your Maple version, such as Maple 2019, Maple 18, etc.?

@Mohamed Abohamer Thanks for the reformatting. I am working on it currently.