A catenary cable is one that is hung between two points not in the same vertical line. As depicted in Figure a, it is subject to no loads other than its own weight. Thus, its weight (N/m) acts as a uniform load per unit length along the cable. A free-body diagram of a section AB is depicted in Figure b, where T_A and T_B are the tension forces at the end. Based on horizontal and vertical force balances, the following differential equation model of the cable can be derived:

Calculus can be employed to solved this equation for the height y of the cable as a function of distance x,

Where the hyperbolic cosine can be computed by cosh x = ? (e^x + e ^?x)

Use a numerical method to calculate a value for the parameter T_A given values for the parameter w = 12 and y₀ = 6, such that the cable has a height of y = 15 at x = 50.

d?y dx? dr I+ dx