A classic task from surveying that is unfortunately no longer taught in our GPS age and is worth remembering:

A hiker has lost his way and wants to know where he is. He has a map, a compass, paper, pen and calculator in his bag. From his position he sees three distant objects from left to right: a radio mast F, a chimney S and a church tower K. He also finds these objects on his map. Using his compass he aims at the three objects and measures the angles at which the distances FS and SK appear: angle for FS=alpha, angle for SK=beta. The hiker also manages to get the approximate coordinates of the three objects from the map to scale: F=(xf;yf), S=(xs;ys) and K=(xk;yk).
Question:
What are the hiker's coordinates?

Given a conventionally labeled triangle ABC with the two sides a=3 and b=4, c is not given. What is the side length of the largest inscribed square whose "base" lies on the triangle side c?

The usual ODE must be solved:
y´´*(y^3-y)+y´^2 *(y^2+1)=0
"Dangerous places" of the definition domain must be described: Where are the general solution y(x) and its derivatives continuous?

A task that was famous at the time is worth remembering:

If for whole numbers x and y the number N = (x^2+y^2)/(1+x*y) is a positive whole number, then it is also a square number.

It can be proven that the converse is also true. Therefore, here is the task:

If N is a square number, then the Diophantine equation has solutions. Solutions must be calculated for N = 9, 49 and 729.

While cleaning up old documents, I found the following Diophantine equation and its solution. I tried to solve it using Maple´s "isolve" but it didn't work. Please give me some advice.
Equation:
x^2 - 12*x*y + 6*y^2 + 4*x + 12*y - 3=0