1. The problem statement, all variables and given/known data I am trying to find an equation for a free hanging chain of mass m and length L. The chain is hanging vertically downwards where x is measured vertically upwards from the free end of the chain and y is measured horizontally. 2. Relevant equations I derived this differential equation for the chains motion, (1/g)(second derivative of y with respect to t) = (derivative of y with respect to x) + x(second derivative of y with respect to x) Trial solution that was given is, y = u(x)cos(ωt) where x ⇒ ξ and ξ = √x and u(x) ⇒ s(ξ) 3. The attempt at a solution By putting y into the equation above I get, -((ω^2)/g)u(x) = u'(x) + xu''(x) where the ' means the derivative with respect to x I think its implied that -((ω^2)/g)s(ξ) = s'(ξ) + xs''(ξ) ... equation 1 After changing the variable to s(ξ) i'm suppose to get a Bessel's equation of order zero I think. so from the change of variables, u(x) ⇒ s(ξ) s'(ξ) ⇒ u'(√x) = (u'(√x))/(2√x) right? and from quotient rule I got, s''(ξ) ⇒ u''(√x) = (u''(√x) - u''(√x)(1/√x)) / (4x) plugging these into equation 1 I get, (x^2)s''(ξ) + x^(3/2)s'(ξ) + (((ω^2)x^2)/g)s(ξ) = 0 but the bessel equation looks like (x^2)s''(ξ) + (x)s'(ξ) + constant(x^2)s(ξ) = 0 which has x instead of x^(3/2)? Not sure what I did wrong?