Yes, it seems the explicit solution has chosen y(0) = sqrt(2)/2 rather than an arbitrary value.

Download dsolve_constants.mw

Replace 
 

a[i, j] := -add(a[i, k], k = 1 .. N, k<>i) 

with

a[i, j] := -(add(a[i, k], k = 1 .. N) - a[i, i])

That is, just add all the terms and then remove the term you don't want.

Here are two ways:

with(Degrees):
evalf(17*sind(34)/sind(115));

or

with(Units):
evalf(17*sin(34*Unit(degree))/sin(115*Unit(degree)));

(You can enter the units above with the Units palette.)

For degree minute seconds I think you will have to do the manipulations yourself

As the error message says, the 2nd argument to eval should be an equation, not just the Matrix, i.e., eval(O, something = M)

Here the derivative of X__2 in terms of X__1 and its derivatives. I don't think it will ever be simple.

compact_derivatives.mw

If I run your worksheet, I get different answers than shown; u[3] etc now have their correct values.

As a separate issue, then you set a value for (u[i])[1], which doesn't make sense - now you make tables u[2] etc that are different from the table u, and you undo the assignments you made before.

Download table.mw

"A system of two first order differential equations produces a direction field plot, provided the system is determined to be autonomous. In addition, a single first order differential equation produces a direction field (as it can always be mapped to a system of two first order autonomous differential equations). A system is determined to be autonomous when all terms and factors, other than the differential, are free of the independent variable. "

The syntax is correct, just there are no solutions in this range. Works for -2..2

Download fsolve.mw

Seems like overkill, but here is one way. As for why, ...
(Note it is complexfreqvar that was s, not statevariable.)

Download dynamic.mw

sol := solve({eq1, eq2, eq3, x + y + z = 1}, {a, b, c, z}, explicit);

I have f(0, 12) taking 44 s, so you might get a result for f(0,30) in managable time.

For future reference please upload your worksheet using the green up-arrow in the Mapleprimes editor.

Download sum.mw

Maple likes the expanded form and one has to "fake it" in order to see exactly what you want. These sorts of manipulations can usually be done, but are not easy and IMO not worth the effort. Maple likes two fewer parentheses - isn't that good?

Download content.mw

Your code uses u in three different ways: as a procedure, as indexed variables, and as a 2-dimensional Array(1..40, 0..4). These should be using different symbols. By the time you enter eval_derivatives, u is the Array, and the other uses are gone. Then you use u[0], which means the zeroth row of the Array. But the rows start at 1, so you get the error message.

Also, I see later in eval_derivatives, you use T1i[i, k - j], when T1i is a 1D Array.

By default, Maple assumes generic values of the coefficients. For special values, one needs to work harder. Here is one way to do it.

Edit: solve([eq1, eq2], [A__1, A__2, w], explicit) works also.

SolveTools:-PolynomialSystem([eq1, eq2], [A__1, A__2, w], explicit)

Download solve.mw

I rewrote this as a table. You don't say if you want all possible p and q sets for each expression, or just one. The following shows both possibilities. As I told you before, you were not properly detecting duplicates because the same expression was appearing in slightly different forms. evalc is enough to make a canonical form here; simplify is more obvious but too expensive.

There are various changes that could be make depending on how you want the output.

Download generate_solution3.mw