Replace 1 with 1.0001 in the square root:

plots:-shadebetween(2,1/sqrt(1.0001-x^2), x=0..sqrt(3)/2);

That looks like a bug to me.  I suggest that you report it.

In the meanwhile, you may get the desired plot by "cheating" as follows.  Exchange the x and y axes temporarilty, and plot whatever you want "sideways":

plot(
  [[2, x, x=0..sqrt(3)/2],
   [1/sqrt(1-x^2), x, x=0..sqrt(3)/2]
  ], color=[red,blue], thickness=3, filled=[color=yellow]);    p := %:

Then apply plottools:-transform() to restore the x and y axes to their standard positions:

plottools:-transform((x,y)->[y,x])(p);

Producing a plot of that motion is quite trivial in Maple, and it is quite likely that that several people will show you how.  However, since you are taking a course in advanced dynamics, I think you will benefit from figuring this out in your head.  I suggest that you begin with simpler problems.

1. What does the motion
   
   r(t) = cos(ωt) i + sin(ωt) j
   
   look like?  If you don't see that, then I am afraid that this is not quite the right time for advanced dynamics.
   
   
2. What does the motion
   
   r(t) = (a cos(ωt))i + (b sin(ωt))j
   
   look like?  This is only slightly different from the previous one.
   
   
3. Now you should be ready to answer your original question.  None of these requires help from Maple.

de := diff(y(x),x,x) + y(x) = sin(x);

eval(de, y(x)=sin(x));

I see that you have posted an answer to your own question.  In the meantime I came up with my own solution which, in retrospect, is very similar to yours, but perhaps a little more organized.  Here it is  mw.mw for whatever it's worth.

You have posted an image of your expression.  It would have been helpful to post a real expression which I could copy and paste on my Maple worksheet.  Not having access to that expressions, I will attempt to sketch the idea of a solution.

1. Generate a sequence of coefficients, such as u*u_xx, etc., which you are interested in:
       C := seq(seq(seq(u^k*diff(u(x,y), [x$i], [y$j]), i=0..2-j), j=0..2), k=0..2);
   Adjust as needed.
2. Let's say your expression is called expr.  Do:
       coeffs(expr, [C], 'terms');
   This will return the the sequence of coefficients you are looking for.  The variable terms will contain the terms from which the coefficients were extracted.

For a similar question, see here.

Short answer:

f := x -> 9*surd(x, 3) + 9/2*x*surd(x,3);

pdsolve is for solving partial differential equations. Yours are ordinary differential equations, so use dsolve instead, as in:

de := diff(f1(t),t) = f2(t),  diff(f2(t),t) = -f2(t);

ic := f1(1)=0, f2(0)=1;

dsolve({de, ic});

plots:-polarplot(tan(t)/cos(t), t=-Pi/3..Pi/3);

with(plots):

x  := [0., .20, .40, .60, .80, 1.0]:
y1 := [0., .20, .40, .60, .80, 1.0]:
y2 := [0., .64, .96, .96, .64, 0.]:

display([
  pointplot(x,y1, connect=true),
  pointplot(x,y2, connect=true)
]);

The title of your post says: "solve this integral equation".  An equation is expected to have an equal sign, as in =, but there is no equal sign in the rest of your post.

I assume that your objective is to evaluate that integral.  If so, the answer is +infinity becasue the integrand behaves like 1/(2*x) as x goes to infinity:

z := 1/((1+x)*(1 + sqrt(x/a)*arccot(1/sqrt(a/x))));

asympt(z, x, 3) assuming a > 0;

Thomas Dean has already pointed out, and here I want to stress, that assigning values to your mathematical variables is not a good worksheet management policy and should be avoided in general.  In your problem, q is a mathematical variable, so seting q := whatever is not a good idea. On the other hand, S is not a part of your mathematics, therefore setting S := { t = 2cm, ... }, as you have done, is just fine.

Here is how to solve your problem without making assignments to your mathematical variables.

and so on...

Here is a complete worksheet: Download mw.mw