@vv No, I meant "statement": Every procedure, arrow expression, or module is required to have a statement, possibly null. I didn't say that that could be any statement. In the case of an arrow expression, the statement must also be an expression, possibly null.

@Carl Love 

Kitonum has correctly pointed out that the tangent line does not exist for the specific example that you used. Nonetheless, what I said still holds: If you want to create multiple objects while overwriting the name, use the unevaluation quotes. 

@Magma Thank you. You've done a great job with the formatting. 

@Magma That technique is extremely inefficient, although it is unfortunately very commonly used, and even appears on some Maple help pages.

Whenever someone asks me to improve the efficiency of their Maple code (as you have done in another thread), that technique is the first thing that I look for.

@Stretto You've asked an excellent and subtle question. The answer is that local declarations and other stuff that may be in a procedure's header are not considered statements. Every procedure (or module or arrow expression) is required to have a statement, which could possibly be a null statement.

Also, it is a commonly held---yet erroneous---belief that statements need to be terminated with a semicolon or colon. This is even expressed on some Maple help pages. The truth is that (just like common English punctuation), those characters are used to separate statements.

@Stretto If S is your series, do

eval(S, sin= 0);

To understand how and why this syntax works: Any number, such as the 0 in this case, can be used as if it were a constant function.

Please supply your procedure in plaintext form. And it's about time that you learned to format and indent them properly. So instead of using some automatic plaintext conversion, please retype your procedure as indented plaintext. Why should I need to do that?

@Kitonum Yes, I was just trying to emphasize that the default definition of series convergence seems to be different between Maple and Mathematica, and that Maple's is the standard, Calculus II, definition.

I have no problem with there being an alternate definition named formal, or even multiple alternate definitions, as long as it's clear which definition is being used. 

You can't make a variable independent just by declaring it to be so. The equation that you show can be solved for r in terms of phi and Z. Doesn't that violate r's supposed independence?

Why can't you adapt the Answer given for your previous extremely similar Question to this new function? What trouble are you having with that? Show an attempt.

And why can't you stop posting poorly formatted output, like I asked before?

@Magma Yeah, of course, I've already read it a few times. Still thinking about it. 

@acer The fact that formal works as you show suggests that Mathematica's default definition of series convergence is nonstandard. Perhaps we should call it A New Kind of Convergence.

@AzU Thank you for reposting the question. Here is my dsolve(..., numeric) solution. I've tried several other solution methods that confirm that the plot is correct. At the bottom of my worksheet, I show how to get a Taylor-polynomial solution. But getting a polynomial solution accurate over the interval 0..1000 will take a huge number of terms.
 

Download IVP.mw

@Christian Wolinski You are conflating two distinct definitions of constant (noun):

1. A symbol which for the purposes of a specific problem or solution process is not to be considered a variable or an independent variable.
2. A symbol representing a specific number or other fixed, well-defined mathematical entity.

When you speak of redeclaring constants in the context of a solver, you are using the first definiton. However, the predefined (yet modifiable) global constants refers to the second definition. There's no need to mix the two. Maple lets you create your own environment variables to your heart's content. So, if you want to use _EnvConstants for the solver purpose that you describe (which I agree is potentially useful), you may do so. 

@ondrejkubu The only part of that process that doesn't seem totally automatic to me is choosing which independent variable to declare in the identity clause. Let me know if you have any trouble with that.