member(2,{$1..7}); true

See ?SurfaceOfRevolution and ?SurfaceOfRevolutionTutor in Student:-Calculus1 package.

The ?simplify help page explains that behaviour as follows:

• Note that when the symbolic option is specified, any branch of a multi-valued function can be chosen during the simplification process. The result of such an operation is in general not valid over the whole complex plane and can lead to incorrect results when the expressions are understood to represent analytical functions.

That depends on P2. In general, smallest (positive) root might not exist at all - for example, for P2(x)=sin(1/x). If you mean smallest positive root, if it exists, it can be found, for example, by division the segment in 2 parts. For example, first find x0:=fsolve(P2), then (if it is positive) try x1:=fsolve(P2,0..x0/2). If it exists, try fsolve(P2,0..x1/2) etc.; if not - try fsolve(P2,x0/2..x1) etc. That can miss some roots though, because sometimes fsolve can't find a root on an interval while a root exists. If possible, plot P2, determine a good approximation for the smallest root visually and enter it in fsolve as an initial value.

If you typed ?fit and read the help page, you wouldn't have to ask this question,

Statistics:-Fit(a+b*sin(Pi*x/6-c),
Vector([$1..12]),
Vector([43.2,47.2,53.7,60.9,70.5,82.1,88.0,86.0,77.4,66.0,51.5,44.1]),
x);

  64.2166666666667823

                                   Pi x
         + 22.1675851546618503 sin(---- - 21.0159675657299943)
                                    6

Here is an example,

b:=plot3d(iL_RMS_func_Ue_const, 0..2, 0..1, axes=normal, 
title="RMS current through the inductor with Ue constant and Ua variable", 
labels=["Ia_IaLGmax", "Ua_Ue", "iL_RMS_Ue_const"]):

a:=plot(x^2/4,x=0..2,y=0..1,numpoints=25):

c:=map(x->[op(x),op([1,3],b)[round(x[1]*12)+1,round(x[2]*24)+1]],op([1,1],a)): 

plots[display](b,PLOT3D(CURVES(c),COLOUR(RGB,1.,0.,0.)));

Note the use of numpoints=25 in the curve plotting command - it matches the array size in the surface plotting command.

1. (D@@2)(nu)(0)=0 in Maple input, ν''(0)=0 in 2-D math. 2. Delta function in Maple is Dirac. One can use it the same way as any other function. For example, dsolve(diff(y(x),x)=Dirac(x-a)); y(x) = Heaviside(x - a) + _C1

I don't have Matlab on this computer, so I can't check it, but evalM in Matlab package seems doing that.

The procedure that I described, assumes that x0<0. It shouldn't work with x0=0 for several different reasons including non-uniqueness of the solution. Also, method=dverk78 was just an example - I used it just because fsolve crashed Maple with rkf45 in that example. Any other method can be used, and many other methods are better. What is x1 in your formulation of the problem here? Is it the same as x0 in earlier posting? Anyway, if it is 0, then no need in this procedure - it follows immediately from Eq, that f(0)=ug that is a usual initial condition - not 'implicit' or 'mixed'. Why are you writing f(x=0) and f(x=x1)? - it is not only nonstandard, but it is just wrong - the usual notation is f(0) and f(x1).

You need a computer projector with brightness 5,000 lumen or more. Then Maple can be demonstrated on screen right from a notebook. Maple's worksheet can be organized in collapsing subsections, so that they can be opened after clicking on a triangle at the beginning of a subsection. Also, tables produce good visual effects. I mean new tables - like in HTML, with a possibility of entering text etc. in few columns. Solving differential equations (entered with y' notation) by rightclicking on them and selecting "solve DE" in the context menu looks very effective. In general, context menus can be used for effects like that. Certainly, animations look good.

Click right mouse button on a 3d-plot, click on Export As, then click Persistence of Vision (POV), name the file and save it. Sorry, but I prefer not to reply by email.

The variables in Maple procedures can be either local (defined only inside of a procedure), or global (defined both inside and outside of the procedure). Usually a procedure starts with a declaration of local variables - something like

inductor_current_RMS_AVG_p:=proc(Ue,Ua,Ia,L,T,art)
local delta_Ia_con_p, inititial_inductor_current_con_p, 
initial_inductor_current_p, inductor_current_increase_p, 
inductor_current_decrease_p, inductor_current_p;

Without such a declaration, Maple issues warnings. The procedure still should work. The problem is not in that - the problem is that IaLG_boundary_condition is not defined. Assigning it to anything - something like

IaLG_boundary_condition:=200;

would make the procedure working. None of the eval are necessary.

First, S is not a set - it is a sequence. Second, assignment S:=S,i^2 just adds i^2 to the sequence S. There are no any null sets in S. The initial assignment S:=NULL assigns S to the null expression sequence. After the first step in the loop S becomes 1. After the second step, S becomes 2-element sequence 1, 4 etc.

It looks as if parts of your procedure after piecewise commands got lost - probably, because of using < signs. It would be great if you could edit it and replace 'less than' signs with &lt;

Yes. In which case one might try dsolve({sysode}) - sometimes maple can do that, and then find constants manually or using solve. If that doesn't work, one might try numerical solution, dsolve({sysode,icsode},numeric). For other options, see ?dsolve.