@mmcdara Thanks for verifying the issue and especially pointing out that this used to work correctly in earlier versions.  I will file a report with Maple Support.

@Kitonum Thanks for confirming the issue and supplying a workaround.  As mmcdara has noted, this is indeed a bug.  I will let Maple Support know.

@nm You don't want u(0,theta)=0.  You want u(0,theta) to be finite.  So we go back to the solution that you have obtained

and apply expand() to get

We see that the last term blows up at the origin. To remove that signularity, we let _C2 = 0:
factor(eval(sol, _C2=0));

which is the correct solution.

@vv Those are good observations.  I wonder how Maple arrives at zeros for the values of those integrals.

@mmcdara Thanks for the idea. This has helped me a lot since I have about 200+ such integrals to calculate.

@Kitonum Thanks for the solution. Actually I thought of the same thing yesterday after I had posted the question.   Mmcdara's answer is more suitable to my needs, but in the absence of that, your solution would have been also usable.

@Thomas Richard As you and Preben have noted, there were a few careless errors in the original worksheet. I have replaced it with a corrected one.

I wouldn't have noticed the typo in theta vs thata since as it happens, the thata was in a conditional branch which is not reached. Thanks to you, and to maplemint, for pointing it out.

As to uses plots, plotttools, I haven't encountered issues in the past by loading plots and plottools globally and then calling their comnands from within procs. I assume the only drawback of doing it my way is that those procs are no longer self-contained / portable, but that's not a concern in this one-off application. If there are other adverse consequences, I would be interested to know.

@Preben Alsholm You are correct in both spotting the errors, and your remedies.

I made a few last minute changes to the worksheet before posting and carelessly did not verify their correctness.  I have uploaded a corrected version now.

@Carl Love Thanks for the compliments. As you have guessed, this worksheet does not present notable technical difficulties.  The one aspect that offered a bit of challenge was that the wheel's spinning is not a simple rigid rotation; its parts move relative to each other as it spins.

@Thomas Richard I didn't know about Maple's intsolve. That's very useful to know.  Thanks!

However, applying intsolve to the integral equation that I showed earlier, says::
Error, (in intsolve) integral equation is not linear.
That's rather odd, because the equation is obviously linear.

It will be good if you could bring this to the attention of the developers.

@Realeboha Maple's "diff" operator takes the traditional derivatives of integer orders. For fractional order derivatives use "fracdiff", as in

alpha := 1/2;
fracdiff(g(t), t, alpha) = g(t)*t^(1-alpha)/GAMMA(2-alpha);

This results in the integral equation

From here on you are on your own since Maple has no facilities for solving integral equations.
 

@acer When you file this, you may want to increase the font size to 64 to make it very obvious that space is being allocated for labels even if they are empty strings.

@acer I see why we are getting different results. In my ~/.mapleinit I have

plots:-setoptions(font=[TIMES,12], labelfont=[TIMES,16]);

If you just copy and paste that line into the worksheet that I provided, you will see that the horizontal axis gets a label of  "x".

But that line should not activate the label, should it?

@acer This is in a worksheet with 1D math input on Linux:

         Maple 2021.1, X86 64 LINUX, May 19 2021, Build ID 1539851

I have attached the worksheet.
 

Download mw.mw
 

@acer As long as you are going to submit a bug report, you might as well add this to your llist:

restart;
plots:-display(
	plot(x^2, x=-1..1),
	plot(1-x^2, x=-1..1),
labels=["",""], overrideoption=true);

According to the documentation, we would expect to see no labels in this plot, but we see that the horizontal axis is still labeled "x".