This is a formulation of a differential equations question $($formulation of a BVP $-$ boundary value problem$).$ Solve in terms of the variables given. A suspension cable of length, L $($metres$)$ and weight per unit length, w $($N$/$m$),$ is under tension, T $($N$)$ and is pin connected on both ends. Calculate the deflection of the cable at all locations along the cable $($i$.$e$.,$ y$($x$)).$

$1.$ Sketch the problem and define the variables

$2.$ Free Body diagram $($assume w is constant over $\backslash$Delta x$)$

$3.$ Conservation Law $($assume small angles: sin $\backslash$theta $\backslash$theta and cos $\backslash$theta $1)$

$4.$ Remove dependence on $\backslash$Delta x

$5.$ Constitutive law to relate $\backslash$theta to y $($assume small angles tan $\backslash$theta $\backslash$theta $)$

Hint: tan $\backslash$theta is equal to slope

$6.$ Combine to form final ODE

$7.$ Setup boundary conditions

$8.$ Solve analytically and calculate the deflection at x $=$L$/2$