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

On my journey of discovery in the Maple world, which is new to me, I have now looked at the linear algebra packages. I am less interested in numerics than in symbolic calculations using matrices. I would like to illustrate this with the following task:

Let A be any regular (n; n) matrix over the real numbers for natural n. The regular (n; n) matrix X that solves the equation

X - A^(-1)*X*A = 0 for each A is to be determined. In this, A^(-1) is the inverse of A. Is there perhaps a symbolic solution for a specifically chosen n?

The solution to this old exercise is known. X is every real multiple of the unit/identity matrix, i.e. the main diagonal is occupied by a constant and all other matrix elements are zero.