@dharr 

... that's exactly what interested me. One of the "strengths" of Zeta/polylog becomes clearly visible. Unfortunately, I'm still too clumsy to plot properly. Instead, I used derive like I did in the old days.

@nm 

... I would like to request an evaluation/transformation of your results with the restriction 0=a<b<2.14. With my limited Maple knowledge, I can't do this, so I'm asking for help. So, I'm not interested in the individual numerical calculations in tabular form.

@nm chini_dgl_a.mw

Happy Easter!

@Alfred_F 

This "unsolvable" equation has a numerical solution for the initial value (1;-1) in the interval [1; 2.700592...]. After a long calculation, I was unable to find a symbolic solution.

@Ronan 

How can point coordinates of the curve be directly retrieved in the plot of the function curve without tabulating (getdata)?

@Ronan 

I didn't know the command "rhs" yet.

@nm 

...exists, but not for every arbitrary initial value. A special solution exists for (sqrt(3);1). There is also another initial value (an incredibly long term).

Quote:

"...For this Abel ode   y'=x+y^3, which is known not to be solvable, Maple hangs on ..."

So the existence and uniqueness theorems of Peano, Picard, and Lindelöf do not apply to this equation if the right-hand side is continuous? But, contrary to the statement in the quote, these very theorems ensure the existence of at least a local solution.

@nm 

Reply to quotes:

"But without using the limit, how else will one find the constants of integration? Are you saying the series must be assumed to have  uniform convergence to use the limit as above?"

Yes.

"I assume Maple also used the limit internally to solve for C2 in this example and give the answer it did. How else could it have found the solution it did otherwise?"

That's probably the case. In such cases, a hint from the software would be helpful.

@nm 

...it should also be noted that uniform convergence of the three summand series must be assumed in the open interval x > 0. Only then can y(t) be assumed to converge uniformly in this interval. And only then is it permissible to calculate y(0) as the sum of the limits of the three summand series.

@nm 

...but that doesn't change the fact that when applying the explicit equation form y = f(t,y), as used in proofs of the existence and uniqueness of solutions, the right-hand side is not continuous/Lipschtz continuous at x = 0. Then Maple's error message should look different, or perhaps a note should appear indicating that a solution was only possible after continuing the solution to the edge of the domain of f(t,y).

Regarding your maple_sol solution, I'd like to know what Maple does with the natural logarithm it contains for x = 0.
The general solution may be correct; I haven't checked it. But classically, no special solution is possible for the initial value at x = 0.

@dharr 

You're right. 2^m is correct instead of 2^(m+1). Thanks for this hint and the tips with the table.

@dharr 

How can coordinates be retrieved from the plot of my "test" file, and how is a table of values ​​created, e.g., from {(m; y) } with y from term (1) from my "test" file for, e.g., m=2...17?

...to a problem that shouldn't be forgotten. There are solutions in the literature that are about 100 years old, and they're quite challenging in theory. However, with today's computational capabilities, it's now possible to come up with new proof ideas and practice mathematical reasoning. Perhaps I'll report on that later ;-) .
BTW: I chose this problem because I want to learn how to use Maple commands from the example solutions so that I can work on specific problems in a targeted manner later. I would be very happy to receive more examples with strongly asymmetrical contours, which do not have to be convex.

...it helped a lot.