With thanks to @sand and @janhardo for their help in solving a variational problem using Maple, I'm presenting a puzzle worth remembering. It was originally published in an IBM competition over 20 years ago under simpler conditions than those presented here, and later posed in a more challenging form in private circles:

A farmer wants to buy a section of a large meadow/pasture for his sheep. He likes a lone, large oak tree on the meadow. This tree is to be the starting and ending point of the fence around the desired section. The fence is exactly 100 meters long. The farmer creates a sketch of the situation. He places the oak tree at the origin of a Cartesian coordinate system with a positive y-axis pointing north and sketches the estimated location of the desired meadow section east of the oak tree. The seller then stipulates that the price per square meter increases linearly eastward by a factor of 1/m³. An agreement is reached, and the farmer now considers the optimal route for the fence to maximize the return on his investment.

The fence route without corners and the value of the purchased meadow must be calculated.

In the attached file, I would like to evaluate the integral according to (16) and the minimum according to (17). I would appreciate your help. The goal is to calculate the coefficients a and b.

Euler_eq.mw

In the attached file test1, two terms are to be compared using the "is" function. Theoretically, these terms are equal. A plot is provided for illustration. However, regardless of which symbol ("equal," "not equal," etc.) is used in "is," the result is always "false." What am I doing wrong?

Download test1.mw

According to the help text in Maple 2024.2, a number of classical integral equations can be solved using "intsolve". The Volterra equation of the first kind, with an upper limit of integration x, is of particular interest. A long time ago, I had to solve a similar equation. This one arose from a model of a real-world process, but instead of x, the upper limit of integration was the function y(x), which I had to calculate. I painstakingly solved it to a good approximation. Is there an algorithm in Maple that can at least calculate an approximate solution, or is a numerical solution, e.g., using Ritz, the only option?

edited: I forgot to upload an example

 test.mw

On my journey of discovery through the Maple world, I now want to try out Maple's convenient features in the complex plane, something that used to be laboriously worked out and demonstrated on the blackboard with chalk. I couldn't find a suitable introduction in the help text. I'm interested in whether a package needs to be loaded and how to handle polynomials, series, and line integrals (I have a reasonable understanding only of the theory).