1. Need with(plots) somewhere to get access to fieldplot3D and display
2. The list of eight initial conditions seems to contain the same four initial conditions twice???
3. In the requested solution range ie DEplot3d(SYS, [x(t), y(t), z(t)], t = -1 .. 1, IC, stepsize = .2, linecolor = blue, axes = boxed) I get no singularities
4. If I change the range of t for which solution curves are to be generated to DEplot3d(SYS, [x(t), y(t), z(t)], t = -10 .. 10, IC, stepsize = .2, linecolor = blue, axes = boxed), then singularites occur. All of these are to the left of -1, which is why they didn't show up at step(3) above

See the attached worsheet, which is slightly reformatted version of your original

diffSing.mw

Read/execute the attached worksheet - it jut illutrates the methods which have already been supplied by Carl/Kitonum

regFit.mw

I'm not 100% sure I understand what you are trying to achieve, but based on my interpretation - try the attached (quick and dirty)

game.mw

I've just executed your worksheet in Maple2015 and all integral evaluate to numerical values: I can't check it in Maple 17 because I don't have that version

By the way it is a bit redundant to use Int() (the inert from of integrate) immediately followed by eval(%), when you could jut use int()

No idea where your displayed error message about Y[9], Y[10] comes from. There is no Y[9], Y[10] in your worksheet - are you sure you executed the restart command??

When I try to run your attached worksheet, I get a different error

Error, (in dsolve/numeric/bvp/convertsys) unable to convert to an explicit first-order system. After a bit of investigation, I came up with the following

1. Eq6 uses the variable "l" (rotten font this - I mean the letter after "k" and before "m"), which has not been assigned a numeric value
2. In your list of boundary conditions you have H(eta)=w, which isn't really a boundary condition! If I change this to H(0)=w, then Maple appears to genrate the necessary procedures for a numeric solution

Generating these procedurs inside a loop and assigning them to p is a bad idea, because the procedures for each value of k, will overwrite those for the previous value of k - suggest you either do these one at a time, or define a k-element, p-array and write the procedures for each k into the appropriate slot

Nothing wrong with Carl's assessment, but his example could also be solved using only lists with something like

restart;
with(plots):
cols:=[red, green, blue, orange, black]:
plotlist:=[ seq(plot(x^k, x = 0..1, color = cols[k]), k = 1..5) ]:
display(plotlist);

Agree with Carl - sooooo many syntax errors

I took a chance, and tried to correct as many as possible - note that in many cases, I'm making an *educated guess* at what you intended, and may have got it wrong somewhere.

With that warning, the attached worksheet supplies plausible-looking results

popMig.mw

If you type ExcelTools[Import]() without giving any arguments, then Maple will fire up an import wizard, which will allow you to navigate to the directory containing the file you want, select the file, and enter other options (sheet, cells etc).

I've just checked both versions (type command with arguments, use the wizard) and both seem to working just fine.

plot3d(1, theta=0..2*Pi, phi=0..Pi, coords=spherical);

Not sure this is going to help much, but here goes

PDEtools:-Solve cannot solve a system, if that system contains an equation in "arbitrary" parameters - in other words one which is completely irrelevant to the variables being solved. For example, as you observe

PDEtools:-Solve(eq3,H[xy])

will work, but add an "arbitrary" equation to this,  such as

PDEtools:-Solve({eq3, a=b+c},H[xy])

and Maple errors.

In the case when you have a system of four equations in six unknowns, and you require a solution for only one variable, then it seems reasonable to suspect that one or more of the equations in the system will be arbitrary.

Note also that when asking for a solution to your system, if you specify four variables, then Maple will generally return a solution, so

PDEtools:-Solve({eq1, eq2, eq3, eq4}, H[xy])

fails, but

PDEtools:-Solve({eq1, eq2, eq3, eq4}, {H[xy],H[tt],H[tx],H[ty]})

succeeds.

I tried a few combinations of four variables and always obtained solutions, although there are 15 possible combinations of four from six variables and I didn't check all of them.

1. Your question is not clear. When you say "what is the meaning of this statement", which statement are you referring to?
2. You should use the big green up-arrow (right-hand end, second row of toolbar) to upload code. Using an image format such as png makes copy/paste almost impossible, and no-one wants to retype your code because it is time-consuming and error-prone
3. If your problem is that fsolve() produces no answers, then (with a bit of work), I came up with the following solutions. There may be more solutions - I can't guarantee it either way

           sol={x = 256.2699202, y = 434.1099130}
           sol={x = 754.9656084, y = 1376.922255}
           sol={x = 1047.750388, y = 1090.143478}
           sol={x = 1594.603214, y = 304.6571162}

You can see my (distinctly experimental) method for doing this in the attached worksheet.

fsolProb.mw

1. I'm assuming that the 'grad' functions refers to the depreated linalg[grad] command, so need to load the (deprecated) linalg package
2. OP's procedure doesn't return anything meaningful, because it defines globals C1, C2, C3, V1, V2, V3 and then assigns values to C[1], C[2}, C[3], V[1], V[2], V[3] - fixed this by assigining to C||j (ie cat the names)
3. To use spacecurves need with(plots)
4. Worksheet now runs (in Maple 2015) and produces plots (see attached), although I'm not sure that these are very interesting (I have no way of knowing)

plotProb.mw

Your requirements appear to be inconsistent:

your third requirement "K:=2 and m:=3" increments the first index, but

your fourth requirement "K:=1 and m:=4" shouldn't increment the first index, but your posted solution does. I'm assuming that this is a typo.

Assuming that it is a typo, try

restart;
genVec:= (x,y) -> [ seq
                             (  [ seq
                                   ( psi[k,j],
                                     k = 1..x
                                   )
                                ],
                                j = 0..y-1
                             )
                          ]:
genVec(1,3);
genVec(1,4);
genVec(2,3);
genVec(2,4);

If this isn't what you want then post requirements in a consistent fashion

1. If you have 9 linear equations with 9 unknowns and numeric coefficients, then I would expect fsolve to breeze through it.
2. If you have 9 non-linear equations with 9 unknowns and numeric coefficients then fsolve might find it a bit harder, but could probably be "persuaded" to work
3. If you have 9 linear equations with 9 unknowns and "algebraic" coefficients, then solve might have difficulty, depending on the algebraic complexity of the coefficients
4. If you have 9 non-linear equations with 9 unknowns and "algebraic" coefficients, then solve will almost certianly have difficulty, particularly if the coefficients are "complicated" algebraic expressions.
5. If you upload a worksheet using the big green up-arrow (right-hand entry in the second row of the toolbar), then you might get some more useful comments

The attached worksheet "undefines" the spherical coordinate system in the Physics package and then "derives" it from first principles. You ought to be able to use it for any other coordinate system, provided that you know the transformation equations.

curCoor.mw

Remember that (according to the help page for the Physics[Vectors] sub-package)

"the Vectors subpackage is designed to work only with cartesian, cylindrical and spherical orthonormal basis and the related systems of coordinates"

so I advise caution if you are going to define your own coordinate system