@okokoabraham No I didn't solve it. Hope someone else can help you. [Edit: see below]

@matmxhu For transendental equations it will always be hard, which I guess is why allsolutions only says "more solutions". The tryhard option doesn't help here either. 

Edit: I guess for these types of second order odes, Sturm-Liouville theory tells you the number of nodes go up by one for each successive eigenfunction, so if you only find half the solutions, you can tell some are missing.

If you assume(n::integer) at the beginning (if that is what you want), then the B..H quantities are much simpler. Still your system is not solving quickly; it's perhaps too hard but I didn't wait that long.

@tomleslie Unfortunately, if that is a bug, then there are many others. In general, one needs to be an experienced user to use "type", "op", "has" and others that require knowing the underlying structure, since in many cases things are not what they seem. For example, from looking at the output of series(exp(x),x) and x/3 one would expect that type(series(exp(x),x),`+`) and has(x,3) would both return true, but they both return false.

@abdulganiy This is more a math question that a Maple question. I don't understand what you want to do.

@Anthrazit I agree that the help implies that your syntax should work, but it is not very specific about whether it works with a table as well as an rtable. (This feature is not available in my version of Maple.)

@nm Yes, these solutions are infinite at the origin, but they satisfy pdetest.

@nm Interesting - that's not f(theta)*g(r) so perhaps a bug, though I suppose it is a sum of those. What about HINT=f(r)*sin(2*theta)?

@one man I took the OP at his word that there were 27, and when my fsolve list had 27 I didn't look much closer. I have now uploaded the worksheet to my answer. From the plots it does look like all are genuine. I didn't check NextZero yet.

@Dr Jean-Michel Collard In another post the OP's PDF file was replaced by a link to researchgate, so I assume the same happened here (I would have instead replaced it by the DOI link). I don't think there is a problem with a link to researchgate.

@Madhukesh J K @tomleslie and @Preben Alsholm have given some code for shooting here that you could adapt to your situation.

@Joe Riel Good point. Of course within a procedure I would normally use: uses  ST=StringTools, rather than any of the others; the only reason to do it here was to see how the procedure body printed.

@Madhukesh J K You did not ask for a particular method before, and as my worksheet shows, Maple's default method works perfectly well. It found one of the second solutions, so it is up to the task of finding the "dual solutions". If you put in close enough approximate solutions, you will get what you want, but at this point your knowledge of the solutions and what they look like is key to moving forward.

@ogunmiloro The message is clear - there are only 13 eqns. You have missed ODE11.

@pik1432 FYI, rather than using j:=I to make j into sqrt(-1), use interface(imaginaryunit=j); This will also change the display from I to j.