Did I solve the hanging cable problem correctly?

[physwizard]

guys i got bored so i decided to try to solve the hanging cable problem - a cable suspended between two arbitrary points. i used the calculus of variations and the functional derivative to minimize the potential energy. i did get the catenary equation, y = a*cosh(x/a+b) + c . for some reason, it does not seem to fit the boundary conditions - y=0 at x=0, and y=a at x=a, i.e. you get imaginary values for the constants. (my origin is at the first point of suspension and here it is assumed that both points are at an equal height above the ground and x-axis is parallel to the ground and passes through the second point of suspension as well). I googled this problem and found that people who solved the problem by keeping the lowest point of the curve as the origin didn't run into the kind of problem i ran into. fundamentally, the choice of origin should not really matter. so did i do something wrong?

 

[tommw]

I would guess your boundary conditions should be y=0 at x=0 and y=0 at x=a, right?
Then you can solve for b=-1/2 and determine the ratio c/a to equal cosh(-0.5).
To get a or c, you'd need another boundary condition.