I have a system with unknown functions: R(x), mu(x), Y(x). I need to find them AND also mu'(x), Y'(x), Y''(x). How do I write it in the code?

I tried something like this:

F := dsolve({cond, sys}, [R(x), mu(x), Y(x)], numeric, output = listprocedure);
Y1 := eval (Y(x), F);
Y2 := x -> diff(Y1(x), x);

It works, but does it work correctly? Y1 is a massive of numbers, is it correct to write "a derivative from a massive of numbers"?

I also tried this:

F := dsolve({cond, sys}, [R(x), mu(x), Y(x), diff(mu(x),x), diff(Y(x),x), diff(Y(x),x$2)], numeric, output = listprocedure);
Y1 := eval (Y(x), F);
Y2 := eval (diff(Y(x),x), F);

But it doesn't work at all and gives me an error on F: 

Error, (in dsolve/numeric/process_input) dependent variables must be functions of a single unknown, the independent variable. Got [diff(Y(x), x), diff(diff(Y(x), x), x), diff(mu(x), x)]

How should I really do it?
3.1.mw
3.2.mw

So I've got a function and try to find an extrema (and there is one on the plot), but Maple gives me empty brackets.

First, earlier it gave me 2 x-coordinates with extrema (both empty), but now (after reloading) it gives one and empty.

Second, how do I set an interval, if I want to look for extrema from x=0 to x=1, for example? I tried extrema (V, {0,1}, x) and (V, {x=0, x=1}, x), but nothing seems to work.

2.mw

I solved a system of differential equations and want to declare an integral with variable upper limit ( T(x) ) from a function I got from the system ( r(x) ). I did so, but Maple doesn't recognise it as a function of one variable, because when I put this T(x) into a system, it gives me an error:

Error, (in dsolve/numeric/process_input) input system must be an ODE system, got independent variables {t, x}

Though there is no t in the system, t was an integration variable when I declared T(x).

Where is the mistake?

2.mw

I need to solve a system of differential-integral equations (see them on the picture). The T(x) function contains an integral with variable upper limit of the wanted function r(x), while the r(x) contains T(x). I don't know how to load the integral into a system, what syntaxix should I use. The boundary conditions and constants are written in the file, as is the error that appears when I try to enter the T(x) function. In the end I need a graphic of r(x) and T(x).

1_с_Y(theta).mw

There are two questions:

1. I have an ODE system solved numerically, but it has a singularity at Pi because of cot(x). I need to know whether the solution goes to infinity at this point or it has some value. I tried the limit operation, but it doesn't seem to work. Am I doing it wrong or is there no way to do such thing? See details in file 1 1.mw.

I know I can evaluate the function at point Pi - 10^(-10), Pi - 10^(-9) etc, and it has really close values there, but still I doubt it is a strong proof of the function not going to infinity.

Also the following "technical" question: why is Maple building the plot from 0 to Pi (or from -Pi to Pi symmetrically), even if I enter different range? For cot(x) it builds the plot normally, not just from -Pi to Pi, althouth cot(x) has singularity at Pi AND at 0, but my plots are somehow broken off at Pi.

2. I have 2 similar procedures and I need the plots of both of them. But first plot is built correctly and for the second there appears an error: "Error, (in plot) procedure expected, as range contains no plotting variable". I tried entering other ranges, including very small ones, but it didn't help. It calculates the values at points of range nicely though. See details in file 2 2.mw.

And one more time "technical" question about warnings in the procedures. I tried typing "local" or "global" before variables, but Maple gives me an error. How do I get rid of the warnings correctly?

I know it is difficult to see into those problems, but I strongly look for help! Thank you for your time!