I want to solve the system of differential equations
sys :=
  diff(x(t,s),t) = y(t,s),
  diff(y(t,s),t) + x(t,s) = 0;
subject to the initial condition
ic := x(0,s) = a(s),
      y(0,s) = b(s);
where a(s) and b(s) are given.

This looks like a system of PDEs but actually it is a system
of ODEs because there are no derivatives with respect to s.
It is easy to obtain the solution by hand:

x(t,s) = b(s)*sin(t) + a(s)*cos(t)
y(t,s) = b(s)*cos(t) - a(s)*sin(t)

I don't know how to get this in Maple, either through dsolve()
or pdsolve().

Actually both dsolve({sys}) and pdsolve({sys}) do return
the correct general solution, however dsolve({sys, ic})
or pdsolve({sys, ic}) produce no output.  Is there a trick
to make the latter work?

Given a list L of n^2 integers, I can fill an nxn matrix through Matrix(n,n,L).  for instance:
n := 4:
L := [$1..n^2]:
A := Matrix(n,n, L);

I don't know how to do this with a 0-based Array.  I wished that as in the case of the Matrix, the command
B := Array(0..n-1, 0..n-1, L);
would assign B[0,0]=1, B[0,1]=2, ..., B[3,3]=16, but it doesn't.

I know how to fill B's entries through doubly nested for-loops but I suspect that there might be a clever way of doing that. If so, then please show me how.

Searching for "typeset but do not execute" in MaplePrimes returns several hits.  The following looks primising:

Clicking on that link, however, leads to a page which has nothing to do with the searched item.  How does one navigate to the correct page?

I want to make sense of the expression

Int(t^2/ln(t)*exp(-t), t=0..infinity);

The denominator vanishes at t=1.  The singularity at t=1 is not integrable.  I want to see whether the integral is defined in the sense of Cauchy principal value.  Thus, I let

K := Int(t^2/ln(t)*exp(-t), t=0..1-a) + Int(t^2/ln(t)*exp(-t), t=1+a..infinity);

and wish to see whether the following limit exists:

limit(K, a=0, right);

Maple cannot evaluate this.  Nor can I.  Alternatively, we may try:

series(K, a=0);

or

series(K, a=0) assuming a>0, a<1;

In both cases Maple says that it is unable to compute the series.

So my question is: Does the Cauchy principal value exist, and can Maple help one to determine that?