Mathematical Model and Derivation

A- Deriving:

    -    Deriving the ODE and solving was the hardest part we had to work with in this project
    -    The mathematicall model for this problem consisted of 2 parts:
            -    The equation of the catenary curve with 3 unknows
            -    Equation representing the length of the curve
    -    To get to this mathematical model, we had to start with some concepts in mind, Least  Potential 
           principle, Lagrange multipliers, and Euler's equation
    -    So  we followed the following steps:
            -    Applied the Least potential energy
            -    Formula of the length of the chain
            -    Apply the lagrange multiplier formula
            -    Euler lagrange Equation
            -    For more details on the steps we did, deriving the equation is the link to the complete derivation
                   and the final step is to integrate and we get to the following:
   
               

B- Solving:

    -    Solving the ODE was not very hard, using some simple substitution and integrating as you can see in
            solving the ODE.