Take a look at the PolynomialIdeals package.  To bring up the help file, with links to example worksheets, type the following at the command prompt, or use the help browser and do a search for PolynomialIdeals.

>?PolynomialIdeals

I see what I missed from your original worksheet.  The delta in your answer should have alerted me that your left hand side of your equation was in terms of the Heaviside function.  The attached worksheet has the left hand side of the diff eq' equal to your Heaviside function.  The steps are the same, althogh the dsolve call has the method=laplace as an option.

Download 221_laplace_dsolve_compare2.mws

Hello Mario.  The worksheet below has an example, where I used your original diff eq' plus the initial conditions and solved it both ways.  The laplace method was done manually.  At the end, as you can see, both of the answers are the same, because the first minus the second come out to 0.  I had a hard time following your worksheet output.  My file was created with the "Classic" interface.

Download 221_laplace_dsolve_compare.mws

>plot(pmf_x(0));

>plot(pmf_x(1));

>plot(pmf_x(2));

>plot(pmf_x(10),thickness=10);

The last line tests your condition for the number being something other than 1, 2, 3.  If you ommit the thickness, you will not be able to see the line on the x-axis.  You can test this by replacing it with the line below:

plot(pmf_x(10));

Take a look at the Student package, Vector Calculus. 

>?Student[VectorCalculus]

>restart:

>with(Student[VectorCalculus]);

>TangentPlane(x^3 - x^2 - y^3 + y^2, x=1, y=1 );

Later edit for part 2:

For a multivariate Taylor series expansion, the third argument specifies the "truncation order'' of the series.  If no value is used, the default is 6.  If you use the defalult, and check the order of your answer, it comes out to 5.  So, for the order to come out to 6, you have to use a value of 7. 

>mtaylor(y^2 / x^3, [x=1,y=-1],6);

>degree(%);

5

>mtaylor(y^2 / x^3, [x=1,y=-1],7);

>degree(%);

6

In Maple, you can do this directly using a command from the StringTools package.

>restart:

>with(StringTools):

>CharacterFrequencies("I have the best dog.");

" " = 4, "." = 1, "I" = 1, "a" = 1, "b" = 1, "d" = 1, "e" = 3, "g" = 1, "h" = 2, "o" = 1, "s" = 1, "t" = 2, "v" = 1

The answer shows 4 spaces, one period, and the rest of letters with the number of occurances for each.

 Maple is case sensitive.  Try the following:

>with(FourierTrigSeries);

and see if that works for you.  I tried the example with spaces in the directory, and it workded, so that is not the problem.

For Windows Use:
Unpack the zip file to a local directory, preferably without spaces in it.
Find the directory in your Maple installation that is called "Users"
Using a text editor, create a file called maple.ini and in that file
paste the line below into it.

libname := libname, `c:/Apps/MySource/Maple/fouriertrigseries`:

The path will be whatever directory you unpacked the zip file into.
Edit the line to reflect your path.  Then save the file as maple.ini
into the Maple/Users directory.

Make sure Maple is not running when you do this.  When Maple starts up, it
checks the maple.ini file in the Users directory.  If you do all of this
correctly, then the examples.mws worksheet should work.  If you want to use
the fouriertrigseries package in your own worksheet, then you can call it
using the "with" command, just like in the example worksheet that was
provided.

>restart: with(plots):

>l1:=plot([[-10,0],[7,0]],thickness=8):

>l2:=plot([[7,0],[7,0]],style=point,symbol=circle,symbolsize=30,color=red):

>l3:=plot([[-10,0],[10,0]],thickness=0):

>display(l1,l2,l3);

The open circle at x=7 indicates that 7 is not included as part of the number

line.  If the circle was a solid, then 7 would be included, indicating an

inequality of x<=7.

>restart:
>f :=(x^2+y^2)^(x^2*y^2);
>ax := 0: bx:= 0: Lim:= 1:
>err := .04:
>delta := 0.5:
>plot3d(f, x=ax-delta..ax+delta, y=bx-delta..bx+delta,
view=Lim-err..Lim+err, axes=boxed,style=surfacecontour,orientation=[-52,90]);
 

With this view, as x and y approach 0, the z-axis approaches 1, for the function above.

See the help files for limit/multi.  You can bring them up by typing the following and hitting the enter key:

?limit/multi
 

For your particular example, there is no solution returned by Maple.  A plot of your function confirms this:

> restart;
> limit((x^2+y^2)^(x^2*y^2), {x = 0, y = 0});

limit((x^2+y^2)^(x^2*y^2), {x = 0, y = 0})

>with(plots,implicitplot):

>implicitplot((x^2+y^2)^(x^2*y^2), x = -10..10, y =-10..10);

There is an example file  that comes with Maple that is called binarytree.mws or binarytree.mw, depending on what version of Maple you are using.  It shows a programming example that you can modify for your own purposes.

The following line will produce a plot:

>plot(Re(evalf[50](f(100,x))), x = 0 .. 1);

The "Re" part is not neccesary for the above plot, although it does not hurt.  Is the generated plot the result that you are looking for?

The unapply command used here in conjunction with the rhs(right hand side) command  takes the output from dsolve and turns it into a "functional" operator.  The help pages have examples that clarify this command.  ?unapply to open up the help page.  Without this, f[0](s)  would return an emplty plot.

Are you sure about your equation?  It simplifies to a straight line, that crosses the x-axis at -1/2.  There is not much you can assume about this.  Maybe you intended to solve for something other than the form of your equation above?