Parabolic cable

import sympy as sp

q, L, H = sp.symbols('q L H',real=True,positive=True)
C1, C2 = sp.symbols('C1 C2')
x = sp.symbols('x',real=True,positive=True)
z = sp.symbols('z',cls=sp.Function,real=True)

ODE = sp.Eq(H * sp.diff(z(x),x,2), -q)

z = sp.dsolve(ODE,z(x)).rhs

eq1 = sp.Eq(z.subs(x,0),0)
eq2 = sp.Eq(z.subs(x,L),0)
sol = sp.solve((eq1,eq2),(C1,C2))
z = z.subs(sol)
display(z)

\[\displaystyle \frac{L q x}{2 H} - \frac{q x^{2}}{2 H}\]

Length based on Taylor approximation

Ltaylor = sp.integrate(sp.nsimplify(1+1/2*sp.diff(z,x)**2),(x,0,L))
display(sp.simplify(Ltaylor))

\[\displaystyle L + \frac{L^{3} q^{2}}{24 H^{2}}\]

Length based on archlength

display(z)
Lexact = sp.simplify(sp.integrate(sp.sqrt(1+sp.diff(z,x)**2),(x,0,L)))
display(Lexact)
Lexact = sp.integrate(sp.sqrt(1+sp.diff(z.subs([(q,5),(L,10),(H,60)]),x)**2),(x,0,10))
display(Lexact)

\[\displaystyle \frac{L q x}{2 H} - \frac{q x^{2}}{2 H}\]

\[\displaystyle \frac{H \operatorname{asinh}{\left(\frac{L q}{2 H} \right)}}{q} + \frac{L \sqrt{4 H^{2} + L^{2} q^{2}}}{4 H}\]

\[\displaystyle 12 \operatorname{asinh}{\left(\frac{5}{12} \right)} + \frac{65}{12}\]

display(Ltaylor.subs([(q,5),(L,10),(H,60)]).evalf())
display(Lexact.subs([(q,5),(L,10),(H,60)]).evalf())

\[\displaystyle 10.2893518518519\]

\[\displaystyle 10.2822479639646\]