@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

@vv 

And the "CompleteSquare" command in particular interests me a lot. I failed at it and chose a different approach. What does f look like after executing it? How does Maple transform a general quadratic form in the variables x, y, and z into the sum of squares? What is the result? This transformation is the trick (I got the result another way). Is there a principal axis transformation in Maple?
The problem can also be solved as a minimum problem.

@Kitonum 

thank You.

@Kitonum How does Maple check the accuracy of satisfying the equation - is it done exactly symbolically or numerically with high precision?

@dharr 

It works. Now I can edit my ancient MC14 files in Maple much more conveniently.

@acer 

... and how is the last plot animated for monotonically increasing t?