@vv 

...for this very instructive solution. It contains nested commands and a procedure with "for" and "if" statements. Back then, the solution involving paper and pen was very strenuous for me.

@janhardo 

This way of thinking and working is completely new to me. So far, I have solved all the tasks myself, down to the last detail. Thanks for the tip. What AI is being used?

@janhardo 

I can decipher the AI ​​procedure. I understand it. I consider deciphering it pointless. There is a much simpler solution, such as the one presented by @vv13957, and this was precisely my previous approach. It is generalizable. Since I'm not interested in solving the problem but rather in the elegance of a solution (and in this case, also in learning the Maple procedure technique), I don't want to go into it in more detail here. The explanations would require a lot of writing, which isn't necessary.
I'm very cautious about AI and sometimes take critical, even questionable, positions:
-AI isn't capable of creativity, but rather applies known knowledge exclusively in previously unknown combinations.
-AI isn't reliably capable of self-diagnosis (in the spirit of the logic classic "all Greeks are liars").

In any case, the procedural solutions found here hit the mark in my view – not only a solution, but also instructive.

@dharr 

...to put the term "birthday child" (regardless of age) in "...", which is common in my home country.

Both solutions (also @vv 13942) are instructive for me as an introduction to procedural technology.

@janhardo 

... the procedure from  vv 13937 helps me to get started with "for" and "if".

@janhardo 

...I was hoping for a Maple procedure as a solution ;-). I'd like to explore such procedures in more detail in the future.

... to your example 4 ;-) .
Determine all intervals on the x-axis where the Lipschitz constant 0<L<1.

The solution to the problem becomes obvious when the ODE is solved in explicit form.

@acer 

...for Your efforts and constructive help. It's especially unusual for me to learn that commands with assigned variables can themselves be used as variables in "outer" commands (nesting). Learned something new again :).

@acer 

Thanks for the help. I'm having trouble understanding the nested command "evalf(collect(simplify(evala(sol)), exp))". What role does "exp" play in the "collect" option?

@sand15 

...ow can the cumbersome numerical terms in the solution y(x) be converted to floating-point numbers?

@sand15 

... A direct hit, that's it.

@Alfred_F 

With my previous post, I wanted to point out that there is a problem with the uniqueness of solutions. A closer look requires some theory:
Without the initial value y'(0)=0, the differential equation only satisfies the assumption of Picard's theorem... via the existence of a general solution. However, the uniqueness of a specific solution under the given "unusual" initial condition is not governed by the theorems known to me. The uniqueness of the solution (given the proven existence of the general solution) is therefore always a special problem in such cases. Since I am unfamiliar with the Maple algorithm for symbolically determining ODE solutions, I suspect this is the reason for the different solution behavior with the "implicit" option.

@janhardo 

According to the problem, the initial value is y'(0)=0. Inserting this into the ODE reveals that the ODE does not allow a computable y(0) for x=0. At x=0, any real ordinate y(0) is possible, since the identity 0=0 is always satisfied.

The attached file shows that the term on the left side of the equation was transformed into an equivalent sum of squares—using pen and paper and without "CompleteSquare." How should "CompleteSquare" be used to display the same result?

Download Diophant2.mw