It would be good if you showed at least some attempt toward solving your own homework problem.  Write down the equations.  Write down the initial and boundary conditions.  These have nothing to do with Maple, so "im totally new to maple" is not a good excuse.  Then try to code and solve your equations in Maple.  If you run into problems with Maple, then say where you are stuck and you will receive help.  But it's not fair to ask someone (not "some1") else to do your homework problem from scratch.

zohre your equations are badly scaled.  You have, instance, 

Have in mind that sinh() and cosh() are more or less like the exponential function.  Does raising e to the power of 100 make any sense in regard to the problem that you wish to analyze?  Probably not.

You need to rescale your variables so that they take values within a resonably "normal" range.  You wouldn't express the distance between Lodon and Paris in micrometers.  You wouldn't express the size of your computer screen in kilometers.  For the same reason, you shouldn't express your x(t) and y(t) in units that lead to strange coefficients that you have shown.

Try, as an exercise, the following rescaled version of your equations:

eq1 := diff(x(t),t) = sinh(x(t)+y(t)) - x(t)*cosh(x(t)+y(t));
eq2 := diff(y(t),t) = sinh(-x(t)+y(t)) - y(t)*cosh(-x(t)+y(t));
dsol := dsolve({eq1, eq2, x(0)=0, y(0)=1}, {x(t), y(t)}, numeric, range=0..10);

Do you know how to do a phase portrait when the nullclines are straght lines?  The case of curved nullclines is no different.

The variable named soln in your worksheet contains the PDE's solution.  You may examine the numerical values of the variables or plot them.  See the help page on pdsolve/numeric for details.  Here are a few examples:

If x(t) = 2 + t + t^2, then x'(t) = 1 + 2*t.  Certainly that isn't equal to y(t) = 1 + 2t + 0.5t^2.  Did you mean to write something else?

Need to be patient when executing the code below; it takes several minutes to complete. There must be ways to speed it up but I haven't bothered.

I have not looked at the details of Maple's series method for solving ODEs, so there may be a way to do what you want.  Consult Maple's documentation on that topic.  My preliminary assessment is that the series method expresses the solution as an infinite sum of nonnegative powers of the independent variable.  Your two differential equations have eta and r in the denominators.  Maple determines, therefore, that the solution cannot be expressed in a series with nonnegative powers.

The transform() function used in Example 1 is defined in the plottools package.  Load it with

with(plottools);

and all will be right.

Just looking through you code, I see the following symbols which are undefined:

dsnumsort()

C1

V1

C2

V2

What do you mean by these?

for i in [3,-2,5,12] do
    print(i^2);
end do;

It's hard to give concrete advice without seeing what you have on your Maple worksheet.  See if placing your cursor at the end of the line and pressing the F3 key helps.  If it doesn't, then post a sample worksheet in which you have pasted your homework problems.  Use the big green up-arrow to upload a worksheet.

There are too many variables involved here to tell.  It depends on how your operating system (not Maple) handles copying/pasting, and how your text editor receives a pasted copy.

My operating system is Linux, and my text editor is vi.  I just copied a block from a Maple worksheet and pasted into vi, and line breaks were preserved.

Sorry for not being able to offer a more useful explanation.

Load the LinearAlgebra package, then have a look at the GenerateMatrix() function.  The help page provides several examples which should interest you.

 Will this do?

As Thomas Richard has pointed out, the images of the equations that you have posted are hard to read.  Well, actually I don't need to read them because I recognize them as the equations of an Euler beam.  If that is so, then the third equation, M(x) = ..., is incorrect.  Fix that first.

The solution of the beam equation generally will involve integrals of the load function q(x).  Your answer does not involve integrals and actually shows q(x) as a plain q.  That answer is correct if q is a constant, that is, the load distribution is uniform.

So here is what you need to do:

1. Using Maple to solve the equation with a constant q is an overkill.  You should be able to solve it with paper and pencil and hardly any effort, with very elementary calculus.  Try that first in order to understand what is going on. 
2. Only after you are done with step 1, see if you can solve the problem in Maple.  Here you will need what you learned in step1, so jumping into Maple without having done that step is not productive.
3. If you have difficulty with finishing your Maple work, post your incomplete Maple worksheet here, and tell us where you need help.

I could show you my solution, but I am afraid that won't help you understand the material.  I want you to learn the subject well, so that the bridge you build won't collapse when I drive over it.