You have written

dWdt := proc (t::float)...

I.e., you request that the argument of dWdt be of type float. But the ScientificErrorAnalysis:-Quantity object is not of type float:

type(ScientificErrorAnalysis:-Quantity(231.928, 0.5e-2),function);
                                 true

type(ScientificErrorAnalysis:-Quantity(231.928, 0.5e-2),float);
                                false

Hence, the error message.

One way:

dw:=a[0] + a[1]*w(q) + a[2]*diff(w(q),q) + a[3]*w(q)^2:
s:=dsolve(dw):
w1:=eval(diff(w(q),q),s):
diff(eval(w1,q=q(x,y,t)),x);

-1/4*tan(1/2*(4*a[0]*a[3]-a[1]^2)^(1/2)*(q(x,y,t)+_C1)/a[2])*(1+tan(1/2*(4*a[0]*a[3]-a[1]^2)^(1/2)*(q(x,y,t)+_C1)/a[2])^2)*(4*a[0]*a[3]-a[1]^2)^(3/2)*diff(q(x,y,t),x)/a[2]^2/a[3]

You may use convert/phaseamp:

ss:=((-omega[0]^2+omega^2)*sin(omega*t)+
cos(omega*t)*omega*lambda)*F[0]/
(omega^4+(lambda^2-2*omega[0]^2)*omega^2+omega[0]^4):

convert(ss,phaseamp,t);


        2       2           4             2      2        4 1/2
  (omega  lambda  + omega[0]  - 2 omega[0]  omega  + omega )

                                      2        2
        cos(omega t - arctan(-omega[0]  + omega , omega lambda)) F[0]

           /       4          2             2       2           4
          /  (omega  + (lambda  - 2 omega[0] ) omega  + omega[0] )
         /

With few exceptions, I keep all my worksheets in Classic mws format, which typically are much smaller (more than an order of magnitude smaller) than mw worksheets with the same text content. So, I find that seaching large collections of these Classic worksheets for a specific word (a command or an option, say) is quite fast and efficient..

It can be written as:

`#mscripts(mi("He"),none(),none(),none(),mn("2"),mn("4"))`;

With the option AllSolutions (in Maple 13):

solve(sin(x)=0,AllSolutions);
                                Pi _Z2

about(_Z2);
Originally _Z2, renamed _Z2~:
  is assumed to be: integer
 

In this case, the workaround is finding the roots of the numerator, that are not roots of the denominator. A first step is:

solve~(({numer,denom})(sin(x)/(1-cos(x))),AllSolutions);
                          {Pi _Z3, 2 Pi _Z4}

about(_Z3,_Z4);
Originally _Z3, renamed _Z3~:
  is assumed to be: integer

Originally _Z4, renamed _Z4~:
  is assumed to be: integer

I am not aware of a command at the user level that manipulates sets symbolically, subtracting {2*Pi*_Z4} from {Pi*_Z3} and yielding {Pi*(2*_Z5+1)} with _Z5 assumed integer (or {Pi*_Z6} with _Z6 assumed odd). But I am confident that something like that exist somewhere in the SolveTools package. Just an issue of looking a bit...

About tracing, it is more of a process of successive approximations, depending on the case. As an initial step, you could start in Maple 13 with something like:

infolevel[all]:=5:
trace(solve):
printlevel:=20:

For finding a solution common to every term in the sum, what is needed is the set of solve applied on each of the operands (i.e. the terms) of the sum. If a common solution exists, then the set has a single element. It can be done by:

solve~({op(x^2-sin(x))},x);#solution
                                 {0}

solve~({op(x^2-sin(x+1))},x);#no solution
                           {-1, 0}

(not in the hasty way that I have written above, as Joe pointed out).

As a workaround you may use in this case:

{map(solve,x^2-sin(x),x)};
                                 {0}

(that the set contains a single element means that it is a solution).

Tracing shows that Maple tries quite different things. But describing what the implemented methods do will take some time. May be that I post on that within a few days if nobody did it before.

I am not sure that "Maple can't guess". I do believe that the ploting algorithm can do better in this case to find a more sensible default vertical range. And in fact, this is what is shown here (my guess is, roughly, that this range is chosen so that an estimate of the derivative does not go beyond some upper bound, in absolute value). 

You are using the document mode, not the worksheet mode, that shows prompts ">" by default. The brackets on the left hand side delimit document blocks.

The error message at the end arises because you are using (1), which is the with statement output, in the eval statement.

It seems unlikely that you coud get a closed form solution of this equation for omega (with solve). So, you should think of numerical solutions.

Anyways, the error message seems a bug.

If the document that you mean is http://www.maplesoft.com/view.aspx?SF=1404/Lesson01.mw, this is a Standard GUI worksheet, not a .mws Classic GUI worksheet. So, it should not be expected opening it in Classic GUI.

Nevertheless, I have got it open in Maple 12.02 Classic GUI, after clicking "yes" to answer the question "Do you wish to continue?". Moreover, it executed without any significant problem.

For instance:

implicitdiff(x*sin(x*y)=x,y,x):
simplify(%,{sin(x*y)=1});
                                - y/x

Or

eq:=x*sin(x*y)=x:
implicitdiff(eq,y,x):
simplify(%,{eq/x});

                                - y/x

A bit shorter:

foldr(algsubs,Xw,w=2*Pi*f,1/w=1/2/Pi/f);

             Pi (sin(2 Pi f) cos(10 Pi f) + sin(8 Pi f))
             -------------------------------------------
                    f (Pi - 4 Pi f) (4 Pi f + Pi)