I recently read something in another forum about solving a simple ordinary differential equation as an initial value problem (file attached). An unexpected solution behavior was interpreted there as a weakness of the software. However, upon closer inspection, the cause is different, and as a software novice, I cannot determine it myself.

Therefore, my question is:

Is there a way, using Maple, to check the Lipschitz condition in the neighborhood of the initial value before starting the solution of an explicit first-order system of differential equations as an initial value problem?

DGL_test.mw

Having started learning Maple about a year ago as a beginner, and now having mastered my preferred area of ​​"ordinary differential equations" fairly independently, I'd like to explore "elliptic curves" in Maple. For practice, I've chosen two problems, for each of which I only know one solution:

y^2 = x^3  -51*x^2 + 867*x - 4792    (17;11)

y^2 - 2*y + 14 = 2*x^3 + 11*x^2 - 29*x - 17    (3;7)

My attempts using commands like "algcurves", "ThueSolve", and "parametrization" have failed. How does one approach such problems in Maple? I'm also particularly interested in the group-theoretically based graphical secant method.

(I'm familiar with the book by Silverman/Tate.)

We are looking for the smallest natural number n with the property that both the digit sum Q(n) of the number n and the digit sum Q(n + 1) of the successor of n are divisible by 5.

During a birthday party, the birthday child realizes: In 1968, I was the same age as the sum of the digits of my birth year. How old will I be now at the end of 2025?

(Please no AI solution)

In the decimal system, specify the smallest natural number k that begins with the digit 7 and has the following additional property:
If you delete the first digit 7 and write it at the end, the newly created number z = (1/3)*k.