I recently solved the following Diophantine equations:
tan(3*pi/x)+4*sin(2*pi/x)-sqrt(x)=0
and
tan(13*pi/x)+4*sin(19*pi/x)-sqrt(x)=0
Unlike my old "pencil and paper" solution, I used Maple to practice with some sub-calculations to get some guesses for the solution. To confirm my guesses, I inserted them into the equations and used "simplify." The result was "zero." Is this "zero" the mathematically exact zero, or does Maple display a very small real number as "zero" after applying "simplify"?

... to solve familiar problems using Maple. The attached exercises "test" and "test1" don't produce the familiar result. What am I doing wrong?

test.mw

test1.mw

For the attached file I ask for help with the plot of sol_1 for x>0.1 .

chini dgl.mw

The attached file "test" asks for the largest power of two contained as a factor in a term. This is easily solved using "simplify." How can this be achieved using factorization?

test.mw

Thanks for the help with the recursion problem, here's a very challenging problem for the weekend ;-). It should be solvable in Maple, at least for examples. I only know of one very theoretical, unfinished solution that I'm still working on:
Given a closed convex curve in the Euclidean plane. It is multiply continuously differentiable, and the curvature is also continuous.
Then there are at least three pairs of points on the curve with the following property:
The tangents at the points of each pair are parallel to each other, and the curvatures are equal.